Farey Fractions

The number associated with my diurnal age today is 27634 and one its properties qualifies for membership in OEIS A119015:

A119015: denominators of "Farey fraction" approximations to \(e\).

I couldn't understand the explanation in the OEIS comments about how these fractions were formed but an investigation of the number in this blog revealed that I'd already dealt with these types of fractions as applied to \(e\) before in a post titled The Mediant on the 5th November 2022. The explanation in that post is much easier to understand but in it I'd only considered the fractions themselves and not their numerators and denominators considered separately, as part of different sequences. The denominator sequence is linked above and here is the corresponding sequence for the numerators:

A119014: numerators of "Farey fraction" approximations to \(e\).

Here are the fractions again (permalink):

5/2, 8/3, 11/4, 19/7, 30/11, 49/18, 68/25, 87/32, 106/39, 193/71, 299/110, 492/181, 685/252, 878/323, 1071/394, 1264/465, 1457/536, 2721/1001, 4178/1537, 6899/2538, 9620/3539, 12341/4540, 15062/5541, 17783/6542, 20504/7543, 23225/8544, 25946/9545, 49171/18089, 75117/27634, 124288/45723, 173459/63812, 222630/81901, 271801/99990, 320972/118079, 370143/136168, 419314/154257, 468485/172346, 517656/190435, 566827/208524, 1084483/398959

Here are the Farey fraction denominators (permalink):

2, 3, 4, 7, 11, 18, 25, 32, 39, 71, 110, 181, 252, 323, 394, 465, 536, 1001, 1537, 2538, 3539, 4540, 5541, 6542, 7543, 8544, 9545, 18089, 27634, 45723, 63812, 81901, 99990, 118079, 136168, 154257, 172346, 190435, 208524, 398959

Here are the Farey fraction numerators (permalink):

5, 8, 11, 19, 30, 49, 68, 87, 106, 193, 299, 492, 685, 878, 1071, 1264, 1457, 2721, 4178, 6899, 9620, 12341, 15062, 17783, 20504, 23225, 25946, 49171, 75117, 124288, 173459, 222630, 271801, 320972, 370143, 419314, 468485, 517656, 566827, 1084483

There are OEIS entries for the numerators and denominators of the Farey fraction approximations to most of the standard constants e.g. \( \pi \). Here are the Farey fractions together with their numerators and denominators for \( \pi \):

The progressive list of approximating Farey fractions is given by (permalink):

[7/2, 10/3, 13/4, 16/5, 19/6, 22/7, 25/8, 47/15, 69/22, 91/29, 113/36, 135/43, 157/50, 179/57, 201/64, 223/71, 245/78, 267/85, 289/92, 311/99, 333/106, 355/113, 688/219, 1043/332, 1398/445, 1753/558, 2108/671, 2463/784, 2818/897, 3173/1010, 3528/1123, 3883/1236, 4238/1349, 4593/1462, 4948/1575, 5303/1688, 5658/1801, 6013/1914, 6368/2027, 6723/2140, 7078/2253, 7433/2366, 7788/2479, 8143/2592, 8498/2705, 8853/2818, 9208/2931, 9563/3044, 9918/3157, 10273/3270, 10628/3383, 10983/3496, 11338/3609, 11693/3722, 12048/3835, 12403/3948, 12758/4061, 13113/4174, 13468/4287, 13823/4400, 14178/4513, 14533/4626, 14888/4739, 15243/4852, 15598/4965, 15953/5078, 16308/5191, 16663/5304, 17018/5417, 17373/5530, 17728/5643, 18083/5756, 18438/5869, 18793/5982, 19148/6095, 19503/6208, 19858/6321, 20213/6434, 20568/6547, 20923/6660, 21278/6773, 21633/6886, 21988/6999, 22343/7112, 22698/7225, 23053/7338, 23408/7451, 23763/7564, 24118/7677, 24473/7790, 24828/7903, 25183/8016, 25538/8129, 25893/8242, 26248/8355, 26603/8468, 26958/8581, 27313/8694, 27668/8807, 28023/8920, 28378/9033, 28733/9146, 29088/9259, 29443/9372, 29798/9485, 30153/9598, 30508/9711, 30863/9824, 31218/9937, 31573/10050, 31928/10163, 32283/10276, 32638/10389, 32993/10502, 33348/10615, 33703/10728, 34058/10841, 34413/10954, 34768/11067, 35123/11180, 35478/11293, 35833/11406, 36188/11519, 36543/11632, 36898/11745, 37253/11858, 37608/11971, 37963/12084, 38318/12197, 38673/12310, 39028/12423, 39383/12536, 39738/12649, 40093/12762, 40448/12875, 40803/12988, 41158/13101, 41513/13214, 41868/13327, 42223/13440, 42578/13553, 42933/13666, 43288/13779, 43643/13892, 43998/14005, 44353/14118, 44708/14231, 45063/14344, 45418/14457, 45773/14570, 46128/14683, 46483/14796, 46838/14909, 47193/15022, 47548/15135, 47903/15248, 48258/15361, 48613/15474, 48968/15587, 49323/15700, 49678/15813, 50033/15926, 50388/16039, 50743/16152, 51098/16265, 51453/16378, 51808/16491, 52163/16604, 52518/16717, 52873/16830, 53228/16943, 53583/17056, 53938/17169, 54293/17282, 54648/17395, 55003/17508, 55358/17621, 55713/17734, 56068/17847, 56423/17960, 56778/18073, 57133/18186, 57488/18299, 57843/18412, 58198/18525, 58553/18638, 58908/18751, 59263/18864, 59618/18977, 59973/19090, 60328/19203, 60683/19316, 61038/19429, 61393/19542, 61748/19655, 62103/19768, 62458/19881, 62813/19994, 63168/20107, 63523/20220, 63878/20333, 64233/20446, 64588/20559, 64943/20672, 65298/20785, 65653/20898, 66008/21011, 66363/21124, 66718/21237, 67073/21350, 67428/21463, 67783/21576, 68138/21689, 68493/21802, 68848/21915, 69203/22028, 69558/22141, 69913/22254, 70268/22367, 70623/22480, 70978/22593, 71333/22706, 71688/22819, 72043/22932, 72398/23045, 72753/23158, 73108/23271, 73463/23384, 73818/23497, 74173/23610, 74528/23723, 74883/23836, 75238/23949, 75593/24062, 75948/24175, 76303/24288, 76658/24401, 77013/24514, 77368/24627, 77723/24740, 78078/24853, 78433/24966, 78788/25079, 79143/25192, 79498/25305, 79853/25418, 80208/25531, 80563/25644, 80918/25757, 81273/25870, 81628/25983, 81983/26096, 82338/26209, 82693/26322, 83048/26435, 83403/26548, 83758/26661, 84113/26774, 84468/26887, 84823/27000, 85178/27113, 85533/27226, 85888/27339, 86243/27452, 86598/27565, 86953/27678, 87308/27791, 87663/27904, 88018/28017, 88373/28130, 88728/28243, 89083/28356, 89438/28469, 89793/28582, 90148/28695, 90503/28808, 90858/28921, 91213/29034, 91568/29147, 91923/29260, 92278/29373, 92633/29486, 92988/29599, 93343/29712, 93698/29825, 94053/29938, 94408/30051, 94763/30164, 95118/30277, 95473/30390, 95828/30503, 96183/30616, 96538/30729, 96893/30842, 97248/30955, 97603/31068, 97958/31181, 98313/31294, 98668/31407, 99023/31520, 99378/31633, 99733/31746, 100088/31859, 100443/31972, 100798/32085, 101153/32198, 101508/32311, 101863/32424, 102218/32537, 102573/32650, 102928/32763, 103283/32876, 103638/32989, 103993/33102, 104348/33215, 208341/66317, 312689/99532, 521030/165849, 833719/265381, 1146408/364913, 1980127/630294, 3126535/995207, 4272943/1360120]

Here are the Farey fraction numerators for \( \pi \) (permalink):

7, 10, 13, 16, 19, 22, 25, 47, 69, 91, 113, 135, 157, 179, 201, 223, 245, 267, 289, 311, 333, 355, 688, 1043, 1398, 1753, 2108, 2463, 2818, 3173, 3528, 3883, 4238, 4593, 4948, 5303, 5658, 6013, 6368, 6723, 7078, 7433, 7788, 8143, 8498, 8853, 9208, 9563, 9918, 10273, 10628, 10983, 11338, 11693, 12048, 12403, 12758, 13113, 13468, 13823, 14178, 14533, 14888, 15243, 15598, 15953, 16308, 16663, 17018, 17373, 17728, 18083, 18438, 18793, 19148, 19503, 19858, 20213, 20568, 20923, 21278, 21633, 21988, 22343, 22698, 23053, 23408, 23763, 24118, 24473, 24828, 25183, 25538, 25893, 26248, 26603, 26958, 27313, 27668, 28023, 28378, 28733, 29088, 29443, 29798, 30153, 30508, 30863, 31218, 31573, 31928, 32283, 32638, 32993, 33348, 33703, 34058, 34413, 34768, 35123, 35478, 35833, 36188, 36543, 36898, 37253, 37608, 37963, 38318, 38673, 39028, 39383, 39738, 40093, 40448, 40803, 41158, 41513, 41868, 42223, 42578, 42933, 43288, 43643, 43998, 44353, 44708, 45063, 45418, 45773, 46128, 46483, 46838, 47193, 47548, 47903, 48258, 48613, 48968, 49323, 49678, 50033, 50388, 50743, 51098, 51453, 51808, 52163, 52518, 52873, 53228, 53583, 53938, 54293, 54648, 55003, 55358, 55713, 56068, 56423, 56778, 57133, 57488, 57843, 58198, 58553, 58908, 59263, 59618, 59973, 60328, 60683, 61038, 61393, 61748, 62103, 62458, 62813, 63168, 63523, 63878, 64233, 64588, 64943, 65298, 65653, 66008, 66363, 66718, 67073, 67428, 67783, 68138, 68493, 68848, 69203, 69558, 69913, 70268, 70623, 70978, 71333, 71688, 72043, 72398, 72753, 73108, 73463, 73818, 74173, 74528, 74883, 75238, 75593, 75948, 76303, 76658, 77013, 77368, 77723, 78078, 78433, 78788, 79143, 79498, 79853, 80208, 80563, 80918, 81273, 81628, 81983, 82338, 82693, 83048, 83403, 83758, 84113, 84468, 84823, 85178, 85533, 85888, 86243, 86598, 86953, 87308, 87663, 88018, 88373, 88728, 89083, 89438, 89793, 90148, 90503, 90858, 91213, 91568, 91923, 92278, 92633, 92988, 93343, 93698, 94053, 94408, 94763, 95118, 95473, 95828, 96183, 96538, 96893, 97248, 97603, 97958, 98313, 98668, 99023, 99378, 99733, 100088, 100443, 100798, 101153, 101508, 101863, 102218, 102573, 102928, 103283, 103638, 103993, 104348, 208341, 312689, 521030, 833719, 1146408, 1980127, 3126535, 4272943 (OEIS A097545)

Here are the Farey fraction denominators for \( \pi \) (permalink):

2, 3, 4, 5, 6, 7, 8, 15, 22, 29, 36, 43, 50, 57, 64, 71, 78, 85, 92, 99, 106, 113, 219, 332, 445, 558, 671, 784, 897, 1010, 1123, 1236, 1349, 1462, 1575, 1688, 1801, 1914, 2027, 2140, 2253, 2366, 2479, 2592, 2705, 2818, 2931, 3044, 3157, 3270, 3383, 3496, 3609, 3722, 3835, 3948, 4061, 4174, 4287, 4400, 4513, 4626, 4739, 4852, 4965, 5078, 5191, 5304, 5417, 5530, 5643, 5756, 5869, 5982, 6095, 6208, 6321, 6434, 6547, 6660, 6773, 6886, 6999, 7112, 7225, 7338, 7451, 7564, 7677, 7790, 7903, 8016, 8129, 8242, 8355, 8468, 8581, 8694, 8807, 8920, 9033, 9146, 9259, 9372, 9485, 9598, 9711, 9824, 9937, 10050, 10163, 10276, 10389, 10502, 10615, 10728, 10841, 10954, 11067, 11180, 11293, 11406, 11519, 11632, 11745, 11858, 11971, 12084, 12197, 12310, 12423, 12536, 12649, 12762, 12875, 12988, 13101, 13214, 13327, 13440, 13553, 13666, 13779, 13892, 14005, 14118, 14231, 14344, 14457, 14570, 14683, 14796, 14909, 15022, 15135, 15248, 15361, 15474, 15587, 15700, 15813, 15926, 16039, 16152, 16265, 16378, 16491, 16604, 16717, 16830, 16943, 17056, 17169, 17282, 17395, 17508, 17621, 17734, 17847, 17960, 18073, 18186, 18299, 18412, 18525, 18638, 18751, 18864, 18977, 19090, 19203, 19316, 19429, 19542, 19655, 19768, 19881, 19994, 20107, 20220, 20333, 20446, 20559, 20672, 20785, 20898, 21011, 21124, 21237, 21350, 21463, 21576, 21689, 21802, 21915, 22028, 22141, 22254, 22367, 22480, 22593, 22706, 22819, 22932, 23045, 23158, 23271, 23384, 23497, 23610, 23723, 23836, 23949, 24062, 24175, 24288, 24401, 24514, 24627, 24740, 24853, 24966, 25079, 25192, 25305, 25418, 25531, 25644, 25757, 25870, 25983, 26096, 26209, 26322, 26435, 26548, 26661, 26774, 26887, 27000, 27113, 27226, 27339, 27452, 27565, 27678, 27791, 27904, 28017, 28130, 28243, 28356, 28469, 28582, 28695, 28808, 28921, 29034, 29147, 29260, 29373, 29486, 29599, 29712, 29825, 29938, 30051, 30164, 30277, 30390, 30503, 30616, 30729, 30842, 30955, 31068, 31181, 31294, 31407, 31520, 31633, 31746, 31859, 31972, 32085, 32198, 32311, 32424, 32537, 32650, 32763, 32876, 32989, 33102, 33215, 66317, 99532, 165849, 265381, 364913, 630294, 995207, 1360120 (OEIS A097546)

The progressive list of approximating Farey fractions for \( \sqrt{2} \) is given by (permalink):

3/2, 4/3, 7/5, 10/7, 17/12, 24/17, 41/29, 58/41, 99/70, 140/99, 239/169, 338/239, 577/408, 816/577, 1393/985, 1970/1393, 3363/2378, 4756/3363, 8119/5741, 11482/8119, 19601/13860, 27720/19601, 47321/33461, 66922/47321, 114243/80782, 161564/114243, 275807/195025, 390050/275807, 665857/470832, 941664/665857, 1607521/1136689

Here are the Farey fraction numerators for \( \sqrt{2} \) (permalink):

3, 4, 7, 10, 17, 24, 41, 58, 99, 140, 239, 338, 577, 816, 1393, 1970, 3363, 4756, 8119, 11482, 19601, 27720, 47321, 66922, 114243, 161564, 275807, 390050, 665857, 941664, 1607521 (OEIS A119016)

Here are Farey fraction denominators for \( \sqrt{2} \) (permalink):

2, 3, 5, 7, 12, 17, 29, 41, 70, 99, 169, 239, 408, 577, 985, 1393, 2378, 3363, 5741, 8119, 13860, 19601, 33461, 47321, 80782, 114243, 195025, 275807, 470832, 665857, 1136689 (OEIS A002965)

Other terms that are used in this context are "interleave denominators" and "interleave numerators". The Farey fractions for the golden ratio are the successive ratios of Fibonacci numbers (higher term as numerator and lower term as denominator). Permalink.

Now what about the mathematician to whom these kinds of fractions owe their name. Well, he wasn't really a mathematician but rather a geologist, although he does have an entry in MacTutor biographies of mathematicians. Here is a link. Some details follow:

Born on the 24th of September 1766 in Woburn, Bedfordshire, England

Died on the 6th January 1826 in London, England

Summary: John Farey was an Engish geologist, noted as a mathematician for the Farey sequence which is a listing of the rationals.

A New Take On The Sum Of Digits

Clearly only \(0\) and \(1\) satisfy the equation \(3n=3^n\) but what about the following:$$SOD(3n)=SOD(3^n)$$where SOD stands for the sum of a number's digits. The values of \(n\) that do satisfy are members of OEIS A260906:

A260906: numbers \(n\) such that \(3n\) and \(n^3\) have the same digit sum.

Here are the members of this sequence up to one million:

3, 6, 30, 60, 63, 126, 171, 252, 300, 324, 543, 585, 600, 630, 1260, 1281, 1710, 2520, 2925, 3000, 3240, 5430, 5850, 5946, 6000, 6300, 12600, 12606, 12633, 12810, 14631, 16263, 17100, 21618, 22308, 22971, 24663, 25200, 27633, 28845, 28887, 28965, 29241, 29250, 29625, 29628, 30000, 31752, 32400, 49533, 52308, 54300, 58500, 59460, 60000, 63000, 82962, 89325, 126000, 126060, 126294, 126330, 128100, 132633, 146310, 162630, 171000, 216180, 216558, 223080, 223086, 225333, 229653, 229710, 231633, 233325, 246630, 252000, 258333, 276330, 282333, 283233, 285333, 288450, 288870, 289650, 289656, 292410, 292500, 296250, 296280, 300000, 315333, 317520, 319266, 323328, 324000, 466623, 466653, 493284, 495330, 522333, 523080, 531333, 543000, 546291, 558333, 559665, 585000, 585333, 585972, 594600, 594666, 597333, 600000, 625008, 630000, 636618, 663156, 665325, 666558, 798981, 829620, 888333, 893250

Let's take the last member of the above numbers, 893250:$$ \begin{align} 3 \times 893250 &=2679750 \rightarrow 36 \\ 893250^3 &= 712720211203125000 \rightarrow 36 \end{align}$$One could ask the question as to what values of \(k\) in$$ \text{SOD}(k n) = \text{ SOD}(n^k) $$yield numbers \(n\) that satisfy in the range up to one million? For \(k=2\), there are many and they comprise OEIS A049343:

A049343: numbers \(n\) such that \(2n\) and \(n^2\) have same digit sum.

The initial members are:

0, 2, 9, 11, 18, 20, 29, 38, 45, 47, 90, 99, 101, 110, 119, 144, 146, 180, 182, 189, 198, 200, 245, 290, 299, 335, 344, 351, 362, 369, 380, 398, 450, 452, 459, 461, 468, 470, 479, 488, 495, 497, 639, 729, 794, 839, 848, 900, 929, 954, 990, 999

For example:$$ \begin{align} 2 \times 999 &= 1998 \rightarrow 27\\ 999^2 &=998001 \rightarrow 27 \end{align} $$However for \(k=4\) there are only two values that satisfy (21249 and 212490):$$ \begin{align} 4 \times 21249 &= 84996 \rightarrow 36 \\21249^4 &= 203870311303040001 \rightarrow 36 \\ 4 \times 212490 &= 849960 \rightarrow 36 \\ 212490^4 &= 2038703113030400010000 \rightarrow 36 \end{align} $$In the range up to one million (and probably beyond), no higher values of k satisfy. Oddly for \(k=1/2\), there are 31 values that satisfy in the range up to one million:

4, 324, 400, 1444, 8100, 8464, 26244, 32400, 39204, 40000, 82944, 84100, 142884, 144400, 158404, 202500, 204304, 219024, 220900, 238144, 334084, 422500, 544644, 602176, 627264, 630436, 810000, 842724, 846400, 980100, 984064

For example:$$ \begin{align} \frac{1}{2} \times 984064 &= 492032 \rightarrow 20\\ 984064^{1/2} &= 992 \rightarrow 20 \end{align} $$For \(k=1/3\), there are only three values that satisfy in the range up one million (729, 729000 and 970299). For example:$$ \begin{align} \frac{1}{3} \times 970299 &= 323433 \rightarrow 18\\ 970299^{1/3} &= 99 \rightarrow 18 \end{align} $$So there are fractional values of \(k\), namely 1/2 and 1/3, that satisfy the following as well as the integer values of 1 (trivial), 2, 3 and 4:$$ \text{SOD}(kn) = \text{ SOD}(n^k) $$There are probably other fractional values as well that satisfy but that will do for now.

Prime and Non-Prime Digit Sequence

I've posted extensively about the trajectories formed by repeatedly adding the sum of the odd digits of a number to the number itself while subtracting the sum of the even digits, or vice versa. My most recent post was on this topic was titled Revisiting Odds And Evens. Instead of dealing with odd and even digits, trajectories involving prime and non-prime digit sums can be considered. The prime digits are 2, 3, 5 and 7 while the non-prime digits are 0, 1, 4, 6, 7 and 9.

For example, the number associated with my diurnal age today is 27632 and it has the following trajectory of 3 steps under the prime and non-prime algorithm (permalink):

27632 --> 27640 --> 27639 --> 27636 --> 27636

The trajectory terminates when the number has a sum of primes and sum of non-primes that are equal, here 2 + 7 + 3 = 6 + 6. Numbers such as these are NOT listed in the OEIS but there is a listing for prime numbers with this property. There are 137 such primes in the range up to 40,000.

A371352: prime numbers such that the sum of their prime digits is equal to the sum of their nonprime digits.

The sequence begins (permalink):

167, 211, 541, 617, 761, 853, 1021, 1201, 1423, 1559, 1607, 1973, 2011, 2143, 2341, 2383, 2833, 3467, 3719, 3823, 3917, 4051, 4231, 4637, 4673, 5261, 5443, 5519, 5591, 6473, 6521, 6701, 7193, 7643, 7687, 7867, 8053, 8233, 8677, 9137, 9173, 9371, 9551, 10067, 10243, 10559, 10607, 10739, 10937, 10973, 11471, 11783, 12043, 12263, 12539, 12953, 13187, 13259, 13709, 13781, 13907, 14779, 14797, 15329, 15361, 15581, 15923, 16007, 16223, 17093, 17183, 17497, 17749, 17903, 18371, 18713, 18731, 19037, 19073, 19477, 20011, 20101, 20143, 20341, 20431, 21001, 22613, 23041, 23159, 24103, 25391, 25463, 25601, 25643, 25867, 25913, 25931, 26321, 26501, 28657, 29153, 29531, 30197, 30241, 30467, 30971, 31079, 31259, 31817, 31907, 32083, 32159, 32261, 32401, 32443, 32621, 32803, 33479, 33749, 34607, 34739, 35129, 35291, 36151, 37019, 37181, 37493, 37811, 37879, 37897, 37987, 38711, 38977, 39107, 39251, 39521, 39877

Many number trajectories will end in what I call an "attractor", using my Odds and Evens terminology. This is a number whose sum of prime and non-prime digits are equal. For completeness, here is a list of all the 446 numbers greater than 27630 and less than 40000 with the property that the sum of their prime digits equals the sum of their non-prime digits (permalink):

27636, 27658, 27663, 27685, 27788, 27801, 27810, 27834, 27843, 27856, 27865, 27878, 27887, 27900, 28017, 28033, 28071, 28107, 28125, 28152, 28170, 28215, 28222, 28251, 28303, 28330, 28347, 28374, 28437, 28455, 28473, 28512, 28521, 28545, 28554, 28567, 28576, 28657, 28675, 28701, 28710, 28734, 28743, 28756, 28765, 28778, 28787, 28877, 29007, 29025, 29052, 29070, 29135, 29153, 29205, 29223, 29232, 29250, 29315, 29322, 29351, 29502, 29513, 29520, 29531, 29700, 30036, 30058, 30063, 30085, 30111, 30124, 30142, 30179, 30197, 30214, 30238, 30241, 30283, 30306, 30328, 30339, 30360, 30382, 30393, 30412, 30421, 30445, 30454, 30467, 30476, 30508, 30544, 30580, 30603, 30630, 30647, 30674, 30719, 30746, 30764, 30791, 30805, 30823, 30832, 30850, 30917, 30933, 30971, 31011, 31024, 31042, 31079, 31097, 31101, 31110, 31134, 31143, 31156, 31165, 31178, 31187, 31204, 31226, 31240, 31259, 31262, 31295, 31314, 31338, 31341, 31383, 31402, 31413, 31420, 31431, 31516, 31529, 31561, 31592, 31615, 31622, 31651, 31709, 31718, 31781, 31790, 31817, 31833, 31871, 31907, 31925, 31952, 31970, 32014, 32038, 32041, 32083, 32104, 32126, 32140, 32159, 32162, 32195, 32216, 32229, 32261, 32292, 32308, 32344, 32380, 32401, 32410, 32434, 32443, 32456, 32465, 32478, 32487, 32519, 32546, 32564, 32591, 32612, 32621, 32645, 32654, 32667, 32676, 32748, 32766, 32784, 32803, 32830, 32847, 32874, 32915, 32922, 32951, 33006, 33028, 33039, 33060, 33082, 33093, 33114, 33138, 33141, 33183, 33208, 33244, 33280, 33309, 33318, 33381, 33390, 33411, 33424, 33442, 33479, 33497, 33600, 33749, 33794, 33802, 33813, 33820, 33831, 33903, 33930, 33947, 33974, 34012, 34021, 34045, 34054, 34067, 34076, 34102, 34113, 34120, 34131, 34201, 34210, 34234, 34243, 34256, 34265, 34278, 34287, 34311, 34324, 34342, 34379, 34397, 34405, 34423, 34432, 34450, 34504, 34526, 34540, 34559, 34562, 34595, 34607, 34625, 34652, 34670, 34706, 34728, 34739, 34760, 34782, 34793, 34827, 34872, 34937, 34955, 34973, 35008, 35044, 35080, 35116, 35129, 35161, 35192, 35219, 35246, 35264, 35291, 35404, 35426, 35440, 35459, 35462, 35495, 35549, 35594, 35611, 35624, 35642, 35679, 35697, 35769, 35796, 35800, 35912, 35921, 35945, 35954, 35967, 35976, 36003, 36030, 36047, 36074, 36115, 36122, 36151, 36212, 36221, 36245, 36254, 36267, 36276, 36300, 36407, 36425, 36452, 36470, 36511, 36524, 36542, 36579, 36597, 36627, 36672, 36704, 36726, 36740, 36759, 36762, 36795, 36957, 36975, 37019, 37046, 37064, 37091, 37109, 37118, 37181, 37190, 37248, 37266, 37284, 37349, 37394, 37406, 37428, 37439, 37460, 37482, 37493, 37569, 37596, 37604, 37626, 37640, 37659, 37662, 37695, 37789, 37798, 37811, 37824, 37842, 37879, 37897, 37901, 37910, 37934, 37943, 37956, 37965, 37978, 37987, 38005, 38023, 38032, 38050, 38117, 38133, 38171, 38203, 38230, 38247, 38274, 38302, 38313, 38320, 38331, 38427, 38472, 38500, 38711, 38724, 38742, 38779, 38797, 38977, 39017, 39033, 39071, 39107, 39125, 39152, 39170, 39215, 39222, 39251, 39303, 39330, 39347, 39374, 39437, 39455, 39473, 39512, 39521, 39545, 39554, 39567, 39576, 39657, 39675, 39701, 39710, 39734, 39743, 39756, 39765, 39778, 39787, 39877

Also for completeness, here is a list of all the numbers greater than 27630 and less than 40000 with the property they map back to themselves after two or more repetitions of the prime / non-prime algorithm. In other words, they are "vorticals" each belonging to a "vortex" to use my Odds and Evens nomenclature.

27675, 27684, 27690, 27823, 27829, 27837, 27848, 27855, 27866, 27873, 27884, 27922, 27926, 27933, 27939, 27954, 27955, 27964, 27965, 27972, 27981, 28027, 28030, 28272, 28273, 28277, 28279, 28282, 28287, 28352, 28355, 28356, 28362, 28532, 28536, 28553, 28560, 28723, 28729, 28737, 28748, 28755, 28766, 28773, 28784, 29272, 29276, 29372, 29373, 29377, 29379, 29382, 29387, 29507, 29511, 29512, 29552, 29553, 29557, 29559, 29566, 29567, 29570, 29575, 29580, 29585, 29722, 29726, 29733, 29739, 29754, 29755, 29764, 29765, 29772, 29781, 30003, 30009, 30012, 30016, 30075, 30084, 30090, 30102, 30106, 30135, 30138, 30145, 30148, 30259, 30260, 30277, 30286, 30296, 30342, 30346, 30378, 30383, 30384, 30387, 30391, 30392, 30432, 30436, 30453, 30460, 30562, 30566, 30652, 30655, 30656, 30662, 30782, 30786, 30795, 30801, 30872, 30876, 30927, 30930, 31002, 31006, 31035, 31038, 31045, 31048, 31123, 31129, 31137, 31148, 31155, 31166, 31173, 31184, 31213, 31219, 31276, 31281, 31359, 31360, 31377, 31386, 31396, 31407, 31411, 31412, 31452, 31453, 31457, 31459, 31466, 31467, 31470, 31475, 31480, 31485, 31543, 31549, 31579, 31584, 31672, 31673, 31677, 31679, 31682, 31687, 31763, 31769, 31827, 31830, 32059, 32060, 32077, 32086, 32096, 32113, 32119, 32176, 32181, 32243, 32249, 32279, 32284, 32362, 32366, 32423, 32429, 32437, 32448, 32455, 32466, 32473, 32484, 32582, 32586, 32632, 32636, 32653, 32660, 32793, 32797, 32799, 32806, 32807, 32811, 32852, 32855, 32856, 32862, 32972, 32973, 32977, 32979, 32982, 32987, 33042, 33046, 33078, 33083, 33084, 33087, 33091, 33092, 33159, 33160, 33177, 33186, 33196, 33262, 33266, 33363, 33369, 33402, 33406, 33435, 33438, 33445, 33448, 33583, 33589, 33592, 33596, 33622, 33626, 33633, 33639, 33654, 33655, 33664, 33665, 33672, 33681, 33807, 33811, 33812, 33852, 33853, 33857, 33859, 33866, 33867, 33870, 33875, 33880, 33885, 33952, 33955, 33956, 33962, 34032, 34036, 34053, 34060, 34107, 34111, 34112, 34152, 34153, 34157, 34159, 34166, 34167, 34170, 34175, 34180, 34185, 34223, 34229, 34237, 34248, 34255, 34266, 34273, 34284, 34302, 34306, 34335, 34338, 34345, 34348, 34472, 34476, 34513, 34519, 34576, 34581, 34742, 34746, 34778, 34783, 34784, 34787, 34791, 34792, 35062, 35066, 35143, 35149, 35179, 35184, 35282, 35286, 35383, 35389, 35392, 35396, 35413, 35419, 35476, 35481, 35602, 35606, 35635, 35638, 35645, 35648, 35822, 35826, 35833, 35839, 35854, 35855, 35864, 35865, 35872, 35881, 35932, 35936, 35953, 35960, 36052, 36055, 36056, 36062, 36172, 36173, 36177, 36179, 36182, 36187, 36232, 36236, 36253, 36260, 36322, 36326, 36333, 36339, 36354, 36355, 36364, 36365, 36372, 36381, 36502, 36506, 36535, 36538, 36545, 36548, 36713, 36719, 36776, 36781, 37082, 37086, 37163, 37169, 37293, 37299, 37442, 37446, 37478, 37483, 37484, 37487, 37491, 37492, 37613, 37619, 37676, 37681, 37802, 37806, 37835, 37838, 37845, 37848, 37923, 37929, 37937, 37948, 37955, 37966, 37973, 37984, 38072, 38076, 38127, 38130, 38252, 38255, 38256, 38262, 38307, 38311, 38312, 38352, 38353, 38357, 38359, 38366, 38367, 38370, 38375, 38380, 38385, 38522, 38526, 38533, 38539, 38554, 38555, 38564, 38565, 38572, 38581, 38702, 38706, 38735, 38738, 38745, 38748, 39027, 39030, 39272, 39273, 39277, 39279, 39282, 39287, 39352, 39355, 39356, 39362, 39532, 39536, 39553, 39560, 39723, 39729, 39737, 39748, 39755, 39766, 39773, 39784

Take for example, the first member of the above series of numbers:

27675 --> 27690 --> 27684 --> 27675

Here (27675, 27690, 27684) forms a vortex while the individual members (27675, 27690 and 27684) are vorticals. If a number is not an attractor or a vortical then it will captured either by an attractor or a vortex. For example, 27676 is captured by the vortex just mentioned:

27676 --> 27680 --> 27675 --> 27690 --> 27684 --> 27675

On the other hand, 27659 is captured by the attractor 27658:

27659 --> 27658 --> 27658

I've now included in my multipurpose algorithm the trajectory of a number under the prime and non-prime algorithm. 

Probability of Two Random Integers Being Coprime

What is the probability that two integers, chosen at random, are coprime or relatively prime. In other words, they don't have any factors in common. Let's designate the random integers as \(m\) and \(n\). Let's consider a random prime \(p\). The probability that \(p\) divides \(m\) is \(1/p\) and the probability that \(p\) divides \(n\) is also \(1/p\). Therefore the probability that \(p\) will NOT divide \(m\) or \(n\) is \(1-1/p^2\). We have only considered one prime however, and need to take them all into account. So the probability that \(m\) and \(n\) have no prime factors in common is given by the following formula where \(p_i\) represents the \(p\)-th prime:$$\prod_{i=2}^{\infty} \Big (1-\frac{1}{p_i^2} \Big )= \Big (1-\frac{1}{2^2} \Big ) \Big (1-\frac{1}{3^2} \Big ) \Big ( 1-\frac{1}{5^2} \Big )\dots $$We can evaluate this using the sum of the reciprocals of all the integers squared:$$\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 +\frac{1}{2^2}+\frac{1}{3^2} +\frac{1}{4^2} + \dots$$where \(n\) represents the \(n\)-th integer and where we have:$$\sum_{n=1}^{\infty} \frac{1}{n^2} \times \prod_{i=2}^{\infty} \frac{1}{p_i^2} =1$$The derivation of the above relationship is explained well in this video. However, we know that:$$\sum_{n=1}^{\infty} \frac{1}{n^2} =\zeta(2)=\frac{\pi^2}{6}$$and so$$ \begin{align} \prod_{i=2}^{\infty} \Big (1- \frac{1}{p_i^2} \Big ) &= \frac{6}{\pi^2}\\ &\approx 0.6079271 \dots \end{align}$$Thus the probability that two positive integers chosen at random are coprime is about 61%. It's easy to simulate this on a computer to test out its validity (permalink). See Figure 1.

Figure 1

Fermat Near Misses Revisited

On the 10th of December 2022, almost two years ago now, I made a post titled Fermat Near Misses where I wrote:

I was surprised to find that my diurnal age today, 26914, is part of a triple of numbers that almost satisfies the equation \(x^3+y^3=z^3\) and hence the term "Fermat near misses". There are of course no positive integer solutions to this equation but some solutions only miss out by 1.

For example, the triplet of numbers 9, 10 and 12 satisfy \(x^3+y^3=z^3+1\) where \(x=9\), \(y=10\) and \(z=12\):$$9^3+10^3=12^3+1=1729$$Here 1729 is the famous "taxi cab number". Today, the number associated with my diurnal age is 27630 and it has a property that qualifies it for membership in OEIS A050789:

A050789: consider the Diophantine equation \(x^3+y^3=z^3-1\) or 'Fermat near misses'. The values of \(z\) (see A050787) are arranged in monotonically increasing order. Sequence gives values of \(y\).

So we have:$$17328^3 +27630^3 = 29737^3-1 = 26296107018552$$The sequence begins as follows:

8, 138, 138, 426, 486, 720, 823, 812, 1207, 2292, 2820, 3230, 5610, 5984, 6702, 8675, 11646, 11903, 16806, 17328, 21588, 24965, 27630, 36840, 31212, 37887, 33857, 34566, 49409, 46212, 59022, 66198, 66167, 56503, 69479, 64165, 78244, 89970

Notice how 17328 reappears again, not as an \(x\) value, but as a \(y\) value:$$10866^3+ 17328^3 =18649^3-1 = 6485846213448$$Figure 1 shows the values of \(x\), \(y\) and \(z\) that satisfy these near misses:

Figure 1: permalink

Here is a sorted list of all the initial integers that are involved in these near misses, in other words all the \(x\)'s, \(y\)'s and \(z\)'s together.

6, 8, 9, 71, 135, 138, 144, 172, 236, 242, 372, 426, 486, 505, 566, 575, 577, 720, 729, 791, 812, 823, 904, 1010, 1124, 1207, 1210, 1851, 1938, 1943, 2196, 2292, 2304, 2676, 2820, 3086, 3097, 3230, 3318, 3453, 3753, 4607, 5610, 5625, 5984, 6081, 6560, 6702, 6756, 7251, 7676, 8675, 8703, 8999, 10230, 10866, 11646, 11664, 11903, 12884, 15218, 16806, 16849, 17328, 17384, 18649, 21588, 21609, 24965, 24987, 25765, 27630, 28182, 29196, 29737, 31212, 32882, 33857, 34199, 34566, 36840, 36864, 37513, 37887, 38134, 38239, 41545, 46212, 49409, 49461, 51293, 51762, 54101, 56503, 58462, 59022, 59049, 64165, 66167, 66198, 66465, 68010, 69479, 69709, 71852, 73627, 75263, 78244, 78529, 89970

A Plethora Of Squares

The number associated with my diurnal age today, 27625, has the unique quality that it is the smallest number capable of being expressed as a sum of two squares in exactly eight different ways. Here are the different ways:$$ \begin{align} 20^2+ 165^2 &= 27625\\27^2+ 164^2 &= 27625\\45^2+ 160^2 &=27625\\60^2+ 155^2 &=27625\\83^2+ 144^2 &=27625\\88^2+ 141^2 &= 27625\\101^2+ 132^2 &=27625\\115^2+ 120^2 &=27625 \end{align}$$The number arises from 27625's factorisation where:$$27625 = 5^3 \times 13 \times 17$$To determine the number of ways in which it can be written as the sum of two squares, we add 1 to each index, multiply them together and divide the product by 2. If the product is not even, then we round the result up. In the case of 27625 we have:$$ \begin{align} \frac{(3 +1) \times (1 +1) \times (1+1)}{2} &= \frac{4 \times 2 \times 2}{2} \\ &=8 \end{align}$$This property of 27625 qualifies it for membership in OEIS A016032:

 A016032: least positive integer that is the sum of two squares of positive integers in exactly \(n\) ways.

The initial members of the sequence are:

2, 50, 325, 1105, 8125, 5525, 105625, 27625, 71825, 138125, 5281250, 160225, 1221025, 2442050, 1795625, 801125, 446265625, 2082925, 41259765625, 4005625, 44890625, 30525625, 61051250, 5928325, 303460625, 53955078125, 35409725, 100140625, 1289367675781250

It can be noted that the sequence is not monotonic increasing. For example, 8125 is the smallest number that can be expressed as a sum of two squares in exactly five ways but 5525 is the smallest number that can be expressed as a sum of two squares in exactly six ways.

Now if we square 27625 we have the following factorisation:$$27625^2=5^6 \times 13^2 \times 17^2$$Now this number can be expressed as a sum of two squares in 32 different ways. These are the possible ways:$$ \begin{align} 0^2+ 27625^2 &= 27625^2\\969^2+ 27608^2 &=27625^2\\1175^2+ 27600^2 &=27625^2\\2625^2+ 27500^2 &=27625^2\\3060^2 +27455^2 &=27625^2\\ 3588^2+ 27391^2 &=27625^2\\ 4225^2+27300^2 &=27625^2\\5180^2+ 27135^2 &=27625^2\\5655^2+ 27040^2 &=27625^2\\6600^2+ 26825^2 &=27625^2\\6800^2+ 26775^2 &=27625^2\\7223^2+ 26664^2 &=27625^2\\7735^2+ 26520^2 &=27625^2\\8856^2+ 26167^2 &=27625^2\\9724^2+ 25857^2 &=27625^2\\10220^2+ 25665^2 &=27625^2\\10625^2+ 25500^2 &=27625^2\\10815^2+ 25420^2 &=27625^2\\11700^2+ 25025^2 &=27625^2\\12137^2+ 24816^2 &=27625^2\\13000^2+ 24375^2 &=27625^2\\13847^2+ 23904^2 &=27625^2\\14025^2+ 23800^2 &=27625^2\\14400^2+ 23575^2 &=27625^2\\15620^2 +22785^2 &=27625^2\\16575^2+ 22100^2 &=27625^2\\17340^2+ 21505^2 &=27625^2\\17500^2+ 21375^2 &=27625^2\\18239^2+ 20748^2 &=27625^2\\18600^2+ 20425^2 &=27625^2\\18921^2+ 20128^2 &=27625^2\\19305^2+ 19760^2 &=27625^2 \end{align}$$Once again, we know that there are 32 ways to write this number as a sum of two squares because looking at the indices again we have:$$\begin{align} \frac{(6+1) \times (2+1) \times (2+1)}{2} &= \frac{7 \times 3 \times 3}{2}\\ &= \frac{63}{2}\\ &\rightarrow 32 \text{ rounded up} \end{align}$$Now this property of the square of 27625 qualifies it for membership in OEIS A097244:

A097244: numbers \(n\) that are the hypotenuse of exactly 31 distinct integer-sided right triangles, i.e., \(n^2\) can be written as a sum of two squares in 31 ways.

By 31 ways and not 32 ways is meant that the number can be written a sum of two distinct non-zero numbers in 31 ways. The initial members of this sequence are:

27625, 47125, 55250, 60125, 61625, 66625, 78625, 82875, 86125, 87125, 94250, 99125, 110500, 112625, 118625, 120250, 123250, 129625, 133250, 134125, 141375, 144625, 148625, 155125, 157250, 157625, 164125, 165750, 172250, 174250, 177125

As can be seen, 27625 is the first member of this sequence. 

Stella Octangula

The term "stella octangula" is another name for a "stellated octahedron" such as is shown in Figure 1.

Figure 1: stellated octahedron
Source

I've only made one previous post about stellated polyhedra and that was The Cubohemioctahedron and other Polyhedra back on the 21st July 2019. Here's what Wikipedia had to say about the stellated octahedron:

The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted in Pacioli's De Divina Proportione, 1509.

It is the simplest of five regular polyhedral compounds, and the only regular compound of two tetrahedra. It is also the least dense of the regular polyhedral compounds, having a density of 2.

It can be seen as a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two overlapping equilateral triangles, centrally symmetric to each other, and in the same way the stellated octahedron can be formed from two centrally symmetric overlapping tetrahedra. This can be generalized to any desired amount of higher dimensions; the four-dimensional equivalent construction is the compound of two 5-cells. It can also be seen as one of the stages in the construction of a 3D Koch snowflake, a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure. The first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron.

Associated with this shape are the stella octangula numbers. These are figurate numbers  of the form \(n(2n^2 − 1) \) and they form OEIS A007588:

0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, 5474, 6735, 8176, 9809, 11646, 13699, 15980, 18501, 21274, 24311, 27624, 31225, 35126, 39339, 43876, 48749, 53970, 59551, 65504, 71841, 78574, 85715, 93276, 101269, 109706, 118599, 127960

To quote from Wikipedia again:

There are only two positive square stella octangula numbers. The first is \(1\) and the other is$$9653449 = 3107^2 = (13 × 239)^2$$corresponding to \(n = 1\) and \(n = 169\) respectively. The elliptic curve describing the square stella octangula numbers is:$$m^2=n(2n^2-1)$$Using Geogebra, this curve is shown in Figure 2.

Figure 2 

The stella octangula numbers arise in a parametric family of instances to the crossed ladders problem in which the lengths and heights of the ladders and the height of their crossing point are all integers. In these instances, the ratio between the heights of the two ladders is a stella octangula number.

The shape and these numbers caught my attention because my diurnal age today is 27624 and this number is a member of the OEIS sequence for the case of \(n=24\). These numbers are quite sparse. The previous number was 24311 and the next will be 31225.

Here is a video which shows how to create a stellated octahedron using origami. It was uploaded on the 13th March 2010 but the technique of course is timeless. Figure 3 shows another representation of the shape.

Figure 3: source

Figure 4 shows how the shape can be inscribed in a cube and also illustrates its connection with the hexagon.

Figure 4: source

The site from which Figures 3 and 4 were taken also contains illustrations of nets can be used to construct the models. This approach is easier than the origami method. Instructions for the construction of wire frame models can also be found there.

Figure 5 shows a screenshot from a site that provides a stellated octahedron calculator. Entering a side length of 53 generates a volume of 26318 to the nearest decimal place. However, the \(a\) shown in Figure 5 is a side length whereas the common formula uses edge length \(b\) where \(b=a/2\). The volume of the shape using edge length of \( b \) is \(b^3 \sqrt {2} \).

Figure 5: source

An Interesting Intersection

I've never encountered an intersection of the arithmetic derivative and the totient function before until looking at one of the properties associated with the number 27620, my diurnal age yesterday. This number has an interesting property that qualifies it for membership in OEIS A352332:

A352332: numbers \(k\) for which \(k = \phi(k') + \phi(k'') \), where \( \phi \) is the Euler totient function (A000010), \(k'\) is the arithmetic derivative of \(k\) (A003415) and \(k''\) is the second arithmetic derivative of \(k\) (A068346).

Let's look at 27620 and its arithmetic derivatives (denoted by ' and '') and their totients (denoted by \( \phi \) ). Firstly:$$ \begin{align} 27620 &= 2^2 \times 5 \times 1381\\  (27620)' &=33164 \\(27620)'' &= (33164)'\\ &=33168 \\ \phi(33164) &= 16580 \\ \phi(33168) &= 11040 \\ 27620 &= 16580 + 11040 \end{align} $$Up to 40000, the members of this sequence are 4, 260, 294, 740, 1460, 3140, 3860, 5540, 8420, 10820, 15140, 19940, 21860, 24020, 24260, 27620 and 37460 (permalink). Clearly, numbers satisfying these criteria are few and far between.

This got me thinking about whether there were any numbers that satisfied the criterion that they were simply the sum of their first and second arithmetic derivatives, ignoring the totient function. There are only three numbers in the range up to 100,000 that satisfy. They are 6, 42, 15590 and 47058. Here is the breakdown:$$ \begin{align} 6 &= 5 + 1\\42 &= 41 + 1\\15590 &= 10923 + 466\\47058 &= 47057 + 1 \end{align}$$It can be noted that in all but 15590, the first derivative yields a prime and thus the second derivative is 1.

What if, instead of the totient, we use the sum of the divisors (\(\sigma\)) of the number? In the range up to 100,000, no numbers satisfy but if we use the sum of the PROPER divisors, then a single number qualifies and that is 50 (permalink). Let's look more closely:$$ \begin{align} 50 &= 2 \times 5^2 \\ (50)' &= 45 \\ \sigma(45) -45&= 78 - 45 \\ &=33\\(50)'' &=(45)'\\ &=39\\ \sigma(39)-39 &= 56-39\\ &= 17 \\50 &= 33 +17 \end{align} $$It should be noted that numbers like 50, 15590 and 27620 with the specific properties that have been mentioned in this post retain these properties in bases other than 10. In other words, the properties are base independent.

A Prime To Remember

My diurnal age today is 27617 and the second interesting property that I discovered about this number (my first being that it's prime) was the fact that:$$ \begin{align} 2^1+7^1+6^1+1^1+7^1 &= 23 \\2^2+7^2+6^2+1^2+7^2 &= 139 \\2^3+7^3+6^3+1^3+7^3 &= 911 \end{align}$$The sums of the digits raised to the first, second and third powers (23, 139 and 911) are all prime. Numbers with this property constitute OEIS A176179:

A176179: primes such that the sum of digits, the sum of the squares of digits and the sum of 3rd powers of their digits is also a prime.

In the range up to 40,000, there are 322 numbers that are members of this sequence. They are:

11, 101, 113, 131, 199, 223, 311, 337, 353, 373, 449, 461, 463, 641, 643, 661, 733, 829, 883, 919, 991, 1013, 1031, 1103, 1301, 1439, 1451, 1471, 1493, 1499, 1697, 1741, 1949, 2089, 2111, 2203, 2333, 2441, 2557, 3011, 3037, 3307, 3323, 3347, 3491, 3583, 3637, 3659, 3673, 3853, 4049, 4111, 4139, 4241, 4337, 4373, 4391, 4409, 4421, 4481, 4603, 4663, 4733, 4919, 4931, 5303, 5503, 5527, 5639, 5693, 6043, 6197, 6337, 6359, 6373, 6719, 6733, 6791, 6917, 6971, 7411, 7433, 7691, 8209, 8353, 8803, 8887, 9091, 9109, 9341, 9413, 9419, 9431, 9491, 9901, 9941, 10099, 10103, 10141, 10211, 10301, 10499, 10909, 10949, 11003, 11047, 11113, 11117, 11131, 11159, 11171, 11173, 11243, 11261, 11311, 11317, 11399, 11423, 11443, 11489, 11519, 11621, 11731, 11777, 11821, 11939, 12011, 12101, 12143, 12161, 12211, 12347, 12413, 12437, 12451, 12473, 12541, 12583, 12611, 12743, 12853, 13001, 13049, 13171, 13241, 13313, 13331, 13339, 13421, 13441, 13487, 13711, 13933, 14011, 14051, 14071, 14107, 14143, 14251, 14321, 14327, 14341, 14387, 14431, 14543, 14549, 14723, 14783, 14891, 15241, 15401, 15443, 15823, 16097, 16361, 16631, 17041, 17401, 17483, 17609, 17627, 17977, 18121, 18149, 18211, 18253, 18523, 18743, 19009, 19139, 19319, 19333, 19391, 19403, 19777, 19841, 19913, 20023, 20089, 20333, 20441, 20809, 21011, 21101, 21121, 21143, 21211, 21341, 21347, 21611, 21767, 22003, 22027, 22111, 22229, 22447, 22481, 22483, 23417, 23581, 23741, 24113, 24137, 24151, 24247, 24281, 24317, 24371, 24443, 24821, 24977, 25057, 25183, 25253, 25411, 25453, 25523, 25583, 25589, 26111, 26177, 26399, 26717, 26993, 27143, 27431, 27479, 27527, 27617, 27749, 27947, 28111, 28351, 28513, 28559, 29989, 30011, 30307, 30323, 30347, 30367, 30491, 30637, 30703, 30763, 30853, 30941, 31247, 31333, 31393, 31847, 31991, 32141, 32303, 32411, 32969, 33023, 33113, 33203, 33311, 33331, 33353, 33391, 33533, 33863, 33931, 34019, 34127, 34141, 34211, 34217, 34439, 34703, 34721, 34781, 34871, 35069, 35083, 35281, 35803, 36037, 36073, 36161, 36299, 36307, 36383, 36389, 36697, 36833, 36929, 37003, 38053, 38639, 38693, 39041, 39119, 39133, 39191, 39313, 39443, 39667, 39863

However, 27617 has an even more exclusive claim to fame as shown below where the rightmost numbers represent digit sums. Here it is:$$ \begin{align}27617^1 &= 27617 \rightarrow 23 \\27617^2 &= 762698689 \rightarrow 61\\27617^3 &= 21063449694113 \rightarrow 53\\27617^4 &= 581709290202318721 \rightarrow 67\\27617^5 &=16065065467517436117857 \rightarrow 101\\27617^6 &=443668913016429033266856769 \rightarrow 127\\27617^7 &= 12252804370774720611730783389473 \rightarrow 131 \end{align}$$All the numbers on the right (23, 61, 53, 67, 101, 127 and 131) are prime. 27617 is the smallest prime with this property that qualifies it for membership in OEIS A131748:

A131748: minimum prime that raised to the powers from 1 to \(n\) produces numbers whose sums of digits are also primes.

In the case of 27617, \(n=7\) and the initial members are: 2, 5, 739, 47, 4229, 2803, 27617, 142589, 108271, 2347283, 1108739, 300776929, 300776929, 14674550173, 92799126239

It would appear that 27617 is the only prime that is a member of both these OEIS sequences.

Deceptive Numbers

It's not surprising that I've not come across deceptive numbers before as they are quite light on the ground so to speak. Today, 27613 is the number associated with my diurnal age and it is a deceptive number but the previous such number was 24661 and the next will be 29431. So what constitutes a deceptive number? Here what Numbers Aplenty has to say:

Let us denote with \(R_k\) the repunit \(111\dots 1\) made of \(k\) ones.

It is known that every odd prime \(p\) divides the repunit \(R_{p-1}\).

R. Francis & T. Ray call a composite number \(n\) deceptive if it has the same property, i.e., if it divides the repunit \(R_{n-1}\).

For example, \(91=7 \times 13\) is deceptive because it divides \(R_{90}\).

Francis & Ray have proved that there are infinite deceptive numbers since, if \(n\)  is deceptive, then \(R_n\) is deceptive as well.

Every number greater than 2980 can be written as the sum of deceptive numbers.

Here are the deceptive numbers up to 100,001: 

91, 259, 451, 481, 703, 1729, 2821, 2981, 3367, 4141, 4187, 5461, 6533, 6541, 6601, 7471, 7777, 8149, 8401, 8911, 10001, 11111, 12403, 13981, 14701, 14911, 15211, 15841, 19201, 21931, 22321, 24013, 24661, 27613, 29341, 34133, 34441, 35113, 38503, 41041, 45527, 46657, 48433, 50851, 50881, 52633, 54913, 57181, 63139, 63973, 65311, 66991, 67861, 68101, 75361, 79003, 82513, 83119, 94139, 95161, 97273, 97681, 100001, ...

These numbers comprise OEIS A000864:

 A000864: deceptive nonprimes: composite numbers \(k\) that divide the repunit \(R_{k-1}\).

Some Interesting Constants

There was an interesting tweet by Cliff Pickover today that is shown below as Figure 1.

Figure 1

The URL shown in the tweet leads to the Wikipedia article about it. First and formost however, who was Gelfond? As usual the MacTutor site at St.Andrews provides a brief biography.

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Born: 24th October 1906 in St Petersburg, Russia

Died: 7th November 1968 in Moscow, Russia. 

Summary

Gelfond developed basic techniques in the study of transcendental numbers.

Biography 

Aleksandr Osipovich Gelfond's father was Osip Isaacovich Gelfond who was a physician who also had an interest in philosophy. Gelfond entered Faculty of Physics and Mathematics at Moscow State University in 1924 and completed his undergraduate studies in 1927. He then began research under the supervision of Aleksandr Khinchin and Vyacheslaw Stepanov and completed his postgraduate studies in 1930.

During 1929-30 he taught mathematics at Moscow Technological College but already he had published some important papers: The arithmetic properties of entire functions (1929); Transcendental numbers (1929); and An outline of the history and the present state of the theory of transcendental numbers (1930). The second of these 1929 papers contained the lecture which Gelfond gave to the First All-Union Mathematics Congress held in Kharkov in 1930. These papers by Gelfond represent a major step forward in the study of transcendental numbers. The first of the papers examines the growth of an entire function which assumes integer values for integer arguments. In the second of the 1929 papers Gelfond applied this result to prove that certain numbers are transcendental, so solving a special case of Hilbert's Seventh Problem. We explain some of these ideas below.

In an article he wrote, Gelfond describes the four month visit which he made in 1930 to Germany where he spent time at both Berlin and Göttingen. He was particularly influenced by Hilbert, Siegel and Landau during his visit. After his return to Russia, Gelfond taught mathematics from 1931 at Moscow State University where he held chairs of analysis, theory of numbers and the history of mathematics. From 1933 he also worked in the Mathematical Institute of the Russian Academy of Sciences.

Gelfond developed basic techniques in the study of transcendental numbers, that is numbers that are not the solution of an algebraic equation with rational coefficients. In addition to his important work in the number theory of transcendental numbers, Gelfond made significant contributions to the theory of interpolation and the approximation of functions of a complex variable. He also contributed to the study of differential and integral equations and to the history of mathematics.

In 1934 he proved a special case of his conjecture namely that \(a^x\) is transcendental if \(a\) is algebraic (\(a \neq 0,1 \)) and \(x\) is an irrational, algebraic number. This result is now known as Gelfond's theorem and solved Problem 7 of the list of Hilbert problems. It was solved independently by Schneider. In 1966 Alan Baker proved Gelfond's Conjecture in general. Gelfond's papers in 1933 and 1934, which include his remarkable achievement, are:  

• Gram determinants for stationary series (written jointly with Khinchin) (1933) 

• A necessary and sufficient criterion for the transcendence of a number (1933) 

• Functions that take integer values at the points of a geometric progression (1933) 

• On the seventh problem of D Hilbert (1934) 

Gelfond addressed the Second All-Union Mathematics Congress in Leningrad in 1934) on Transcendental numbers.

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The decimal expansion of Gelfond's Constant is:$$e^{\pi} = 23.14069263277926900572 \dots$$The Wikipedia article states that \(e^{\pi} \) also appears in the volumes of hyperspheres. The volume of an \(n\)-sphere with radius \(R\) is given by:$$V_n(R) = \frac{\pi^{n/2} R^n}{\Gamma (n/2+1)}$$where \(R\) is the gamma function. If we take \(R=1\) we get:$$V_n(1) = \frac{\pi^{n/2} }{\Gamma (n/2+1)}$$Any even dimensional \(2n\)-sphere now gives:$$V_{2n}(1) = \frac{\pi^{n} }{\Gamma (n+1)}$$Summing up all even-dimensional unit sphere volumes and utilizing the series expansion of the exponential function gives:$$ V_{2n}(1) = \sum_{0}^{\infty} \frac{\pi^{n}}{n!}=e^{\pi}$$The Wikipedia article also looks at some related functions and the first of them is Ramanujan's Constant:$$ \begin{align} e^{\pi \sqrt{163}} &= 262537412640768743.99999999999925007259 \dots \\ &\approx 640320^3 + 744 \text{ to within one trillionth}\end{align}$$The article goes on to say that:

This is an application of Heegner numbers, where 163 is the Heegner number in question. This number was discovered in 1859 by the mathematician Charles Hermite. In a 1975 April Fool article in Scientific American magazine, "Mathematical Games" columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name. Ramanujan's constant is also a transcendental number.

There is also the transcendental constant \(e^{\pi}-\pi \) which approximates to:$$e^{\pi}-\pi =19.99909997918947576726 \dots$$The constant \(i^i\) can be evaluated as follows:$$ \begin{align} i^i &= (e^{i\pi /2})^i\\ &= e^{-\pi/2} \\&=(e^{\pi})^{-1/2}\\ &= 0.20787957635076190854 \dots \end{align}$$As for \( \pi^e \), it is not known whether it is transcendental or not.

Following the link above for Charles Hermite yields the following biography. I've included it in full because I believe it's important to promote awareness of the lives of famous mathematicians to whom we owe so much.

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Born: 24 December 1822 in Dieuze, Lorraine, France

Died: 14 January 1901 in Paris, France 

Summary

Charles Hermite's work in the theory of functions includes the application of elliptic functions to the quintic equation. He published the first proof that e is a transcendental number.

Biography

Charles Hermite's father was Ferdinand Hermite and his mother was Madeleine Lallemand. Ferdinand Hermite was a trained engineer and he worked in this capacity in a salt mine near Dieuse. After he married Madeleine he joined in the draper's trade in which her family were involved. However he was an artistic man who always wanted to pursue art as a career. He had his wife look after the draper's business and he took up art. Charles was the sixth of his parents seven children and when he was about seven years old his parents left Dieuse and went to live in Nancy to where the business had moved.

Education was not a high priority for Charles's parents but despite not taking too much personal interest in their children's education, nevertheless they did provide them with good schooling. Charles was something of a worry to his parents for he had a defect in his right foot which meant that he moved around only with difficulty. It was clear that this would present him with problems in finding a career. However he had a happy disposition and bore his disability with a cheerful smile.

Charles attended the Collège de Nancy, then went to Paris where he attended the Collège Henri. In 1840-41 he studied at the Collège Louis-le-Grand where some fifteen years earlier Galois had studied. In fact he was taught mathematics there by Louis Richard who had taught Galois. In some ways Hermite was similar to Galois for he preferred to read papers by Euler, Gauss and Lagrange rather than work for his formal examinations.

If Hermite neglected the studies that he should have concentrated on, he was showing remarkable research ability publishing two papers while at Louis-le-Grand. Also like Galois he was attracted by the problem of solving algebraic equations and one of the two papers attempted to show that the quintic cannot be solved in radicals. That he was unfamiliar with Galois's contributions, despite being at the same school, is not at all surprising since the mathematical community were completely unaware of them at this time. However he might reasonably have known of the contributions of Ruffini and Abel to this question, but apparently he did not.

Again like Galois, Hermite wanted to study at the École Polytechnique and he took a year preparing for the examinations. He was tutored by Catalan in 1841-42 and certainly Hermite fared better than Galois had done for he passed. However it was not a glorious pass for he only attained sixty-eighth place in the ordered list. After one year at the École Polytechnique Hermite was refused the right to continue his studies because of his disability. Clearly this was an unfair decision and some important people were prepared to take up his case and fight for him to have the right to continue as a student at the École Polytechnique. The decision was reversed so that he could continue his studies but strict conditions were imposed. Hermite did not find these conditions acceptable and decided that he would not graduate from the École Polytechnique.

Hermite made friends with important mathematicians at this time and frequently visited Joseph Bertrand. On a personal note this was highly significant for he would marry Joseph Bertrand's sister. More significantly from a mathematical point of view he began corresponding with Jacobi and, despite not shining in his formal education, he was already producing research which was ranking as a leading world-class mathematician. The letters he exchanged with Jacobi show that Hermite had discovered some differential equations satisfied by theta-functions and he was using Fourier series to study them. He had found general solutions to the equations in terms of theta-functions. Hermite may have still been an undergraduate but it is likely that his ideas from around 1843 helped Liouville to his important 1844 results which include the result now known as Liouville's theorem.

After spending five years working towards his degree he took and passed the examinations for the baccalauréat and licence which he was awarded in 1847. In the following year he was appointed to the École Polytechnique, the institution which had tried to prevent him continuing his studies some four years earlier; he was appointed répétiteur and admissions examiner.

Hermite made important contributions to number theory and algebra, orthogonal polynomials, and elliptic functions. He discovered his most significant mathematical results over the ten years following his appointment to the École Polytechnique. In 1848 he proved that doubly periodic functions can be represented as quotients of periodic entire functions. In 1849 Hermite submitted a memoir to the Académie des Sciences which applied Cauchy's residue techniques to doubly periodic functions. Sturm and Cauchy gave a good report on this memoir in 1851 but a priority dispute with Liouville seems to have prevented its publication.

Another topic on which Hermite worked and made important contributions was the theory of quadratic forms. This led him to study invariant theory and he found a reciprocity law relating to binary forms. With his understanding of quadratic forms and invariant theory he created a theory of transformations in 1855. His results on this topic provided connections between number theory, theta functions, and the transformations of abelian functions.

On 14 July 1856 Hermite was elected to the Académie des Sciences. However, despite this achievement, 1856 was a bad year for Hermite for he contracted smallpox. It was Cauchy who, with his strong religious conviction, helped Hermite through the crisis. This had a profound effect on Hermite who, under Cauchy's influence, turned to the Roman Catholic religion. Cauchy was also a very staunch royalist and Hermite was influenced by him to also become a royalist. We made comparisons with Galois earlier on in this article, but with royalist views, Hermite was now completely opposed to the views which the staunch republican Galois had held.

The next mathematical result by Hermite which we must mention is one for which he is rightly famous. Although an algebraic equation of the fifth degree cannot be solved in radicals, a result which was proved by Ruffini and Abel, Hermite showed in 1858 that an algebraic equation of the fifth degree could be solved using elliptic functions. He applied these results to number theory, in particular to class number relations of quadratic forms.

In 1862 Hermite was appointed maître de conférence at the École Polytechnique, a position which had been specially created for him. In the following year he became an examiner there. The year 1869 saw him become a professor when he succeeded Duhamel as professor of analysis both at the École Polytechnique and at the Sorbonne. Hermite resigned his chair at the École Polytechnique in 1876 but continued to hold the chair at the Sorbonne until he retired in 1897. In the 1890s Hermite became much less interested in the new results found by the mathematicians of the next generation.

The 1870s saw Hermite return to problems which had interested him earlier in his career such as problems concerning approximation and interpolation. In 1873 Hermite published the first proof that e is a transcendental number. This is another result for which he is rightly famous. Using method's similar to those of Hermite, Lindemann established in 1882 that π was also transcendental. Many historians of science regret that Hermite, despite doing most of the hard work, failed to use it to prove the result on which would have brought him fame outside the world of mathematics. Hermite is now best known for a number of mathematical entities that bear his name: Hermite polynomials, Hermite's differential equation, Hermite's formula of interpolation and Hermitian matrices.

For Hermite certain areas of mathematics were much more interesting than other areas. Hadamard, who unlike his teacher Hermite worked in all areas of mathematics, spoke of Hermite's dislike for geometry:

[Hermite] had a kind of positive hatred of geometry and once curiously reproached me with having made a geometrical memoir.

Hermite's great love was for analysis and, not surprisingly, he had a great respect for Weierstrass. When Mittag-Leffler arrived in Paris to study with him, Hermite greeted him warmly but said:

You have made a mistake, sir, you should follow Weierstrass's course in Berlin. He is the master of us all.

Poincaré is almost certainly the best known of Hermite's students. He once suggested that Hermite's mind did not proceed in logical fashion. He wrote:

But to call Hermite a logician! Nothing can appear to me more contrary to the truth. Methods always seemed to be born in his mind in some mysterious way.

Hadamard like Poincaré was very interested in the way that mathematics was discovered. He also had this to say about the way that Hermite made his discoveries:

Hermite used to observe [that biology] may be a most useful study even for mathematicians, as hidden and eventually fruitful analogies may appear between processes in both kinds of studies.

Hadamard had great respect for Hermite as a teacher. He said:

I do not think that those who never listened to him can realise how magnificent Hermite's teaching was, overflowing with enthusiasm for science, which seemed to come to life in his voice and whose beauty he never failed to communicate to us, since he felt it so much himself to the very depth of his being.

[Hermite] was making a deep impression on us, not only with his methods and those of Weierstrass, but also with his enthusiasm and love of science; in our brief but fruitful conversations, Hermite loved to direct to me remarks such as: "He who strays from the paths traced by providence crashes." These were the words of a profoundly religious man, but an atheist like me understood them very well, especially when he added at other times: "In mathematics, our role is more of servant than of master." It goes without saying that gradually, as years and my scientific work unfolded, I came to understand more and more deeply the aptness and scope of his words.

Cross, reviewing where 125 letters from Hermite to Mittag-Leffler are reproduced, writes:

So there stands revealed one of the most engaging and influential men in Parisian and French mathematics in the second half of the 19th century, one might even say the central character for the period in which he published, 1842-1901. What radiates from the text is [Hermite's] humility, his Catholicism, his concern for his (very extended) family, his willingness to fight for colleagues whose merit he discerns, and his devotion to family, merit, and principle rather than simple influence.

In terms of his family life Hermite had married Louise Bertrand, Joseph Bertrand's sister. One of their two daughters married Émile Picard. Struik writes:

Hermite lived a retired life, with his family. His working hours were devoted to mathematical research and teaching. His outlook on mathematics was realistic in the Platonic sense: a mathematician, like a naturalist, discovers an outside world, in his case a world of ideas. Hermite, therefore, disliked Cantor's world, in which a new mathematical world was created.

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