A Bit About Betting

I got to thinking about the way that the odds of a horse winning are expressed in betting circles and I sought the help of Gemini 2.0 Flash Experimental to explain it. Here is what it came up with.

Question: If probability of an event happening is given as 2 to 1 on, what does that mean?

When someone says the probability of an event is "2 to 1 on," they're expressing odds in fractional form, specifically "odds on." This means the event is considered more likely to happen than not.   

Here's how to break it down:

2 to 1: This represents the ratio of the likelihood of the event happening versus it not happening. In this case, it means for every 2 times the event is expected to occur, it is expected not to occur 1 time.

"On": This indicates that the first number in the ratio (2 in this case) is larger than the second number (1). This signifies that the event is favored or has a greater than 50% chance of happening.

To convert "2 to 1 on" to a probability:

Add the two numbers in the ratio: 2 + 1 = 3

Divide the first number by the total: 2 / 3 = 0.666...

Therefore, "2 to 1 on" translates to approximately a 66.7% probability or two-thirds chance of the event happening.

Example:

If you're told the odds of a horse winning a race are "2 to 1 on," it implies that out of every three similar races, the horse is expected to win two and lose one.

Key takeaway: "Odds on" always indicate a greater than 50% chance of an event occurring, with the specific probability determined by the ratio provided.

I then asked the following question:

Question: What's an example of a bet that isn't odds on?

You're right to ask for clarification! It's important to understand the difference. A bet that isn't "odds on" means the event is considered less likely to happen than not. This is often referred to as "odds against."

Here's how it works:

The ratio: In "odds against," the second number in the ratio is larger than the first. For example, "3 to 1 against" or "5 to 2 against."

Meaning: This signifies that the event is less favored or has a less than 50% chance of happening.

Example:

Let's say the odds of a particular football team winning their next match are "3 to 1 against."

This means that for every 3 times the team is expected not to win, they are expected to win only 1 time.

To convert this to a probability:

Add the two numbers: 3 + 1 = 4

Divide the second number by the total: 1 / 4 = 0.25

So, "3 to 1 against" translates to a 25% probability of the team winning.

Another example:

If the odds of rolling a 6 on a standard six-sided die are "5 to 1 against," this means that for every 5 times you're expected not to roll a 6, you're expected to roll a 6 only once. This aligns with the actual probability of 1/6 (approximately 16.7%).

In summary:

Odds on: The first number in the ratio is larger (e.g., 2 to 1 on). Indicates a greater than 50% chance.

Odds against: The second number in the ratio is larger (e.g., 3 to 1 against).1 Indicates a less than 50% chance.   

Understanding this distinction is crucial for interpreting betting odds and evaluating the likelihood of different outcomes.

Gapful Numbers

 Numbers Aplenty has this to say about gapful numbers:

L. Colucci calls a number \(n\) of at least 3 digits a gapful number if \(n\)  is divisible by the number formed by the first and last digit of \(n\). For example, 583 is gapful because it is divisible by 53. About the 7.46% of the numbers are gapful.

The smallest Pythagorean triple made of gapful numbers is (120, 160, 200) and the first such primitive triple is (3159, 29440, 29609).

The concept can be extended to other bases. For example, the number 240 is gapful in all the bases from 2 to 10 and the number 54288 is gapful in all the bases from 2 to 16.

The first gapful numbers are 100, 105, 108, 110, 120, 121, 130, 132, 135, 140, 143, 150, 154, 160, 165, 170, 176, 180, 187, 190, 192, 195, 198, 200, 220, 225, 231, 240, 242, 253  

Figure 1 shows the representation of \(54288 = 2^4 \times 3^2 \times 13 \times 29\) in bases from 2 to 16:

Figure 1

Looking at the first and last digits in these different representations we see that:

• Base 2 --> 10 which is 2

• Base 3 --> 20 which is 6

• Base 4 --> 30 which is 12

• Base 5 --> 33 which is 18

• Base 6 --> 10 which is 6

• Base 7 --> 33 which is 24

• Base 8 --> 10 which is 8

• Base 9 --> 80 which is 72

• Base 10 --> 58 which is 58

• Base 11 --> 33 which is 36

• Base 12 --> 20 which is 24

• Base 13 --> 10 which is 13

• Base 14 --> 1a which is 24

• Base 15 --> 13 which is 18

• Base 16 --> d0 which is 13 x 16 = 208

All these numbers (2, 6, 8, 12 etc.) are divisors of 54288 and this why the number is so special.

One way to make these gapful numbers less numerous is to impose an additional criterion. For example, let's require that the sum of the number's digits is equal to the concatenation of the first and last digits. In the range between 1 and 40000, this reduces the numbers satisfying all criteria to 359. The numbers are (permalink):

190, 192, 195, 198, 1090, 1092, 1095, 1098, 1180, 1183, 1185, 1188, 1270, 1272, 1274, 1275, 1278, 1360, 1365, 1368, 1450, 1452, 1455, 1456, 1458, 1540, 1545, 1547, 1548, 1630, 1632, 1635, 1638, 1720, 1725, 1728, 1729, 1810, 1812, 1815, 1818, 1900, 1904, 1905, 1908, 2992, 2997, 10090, 10092, 10094, 10095, 10096, 10098, 10180, 10185, 10188, 10270, 10272, 10275, 10278, 10279, 10360, 10365, 10368, 10450, 10452, 10455, 10458, 10540, 10543, 10545, 10548, 10630, 10632, 10635, 10638, 10720, 10724, 10725, 10727, 10728, 10810, 10812, 10815, 10816, 10818, 10900, 10901, 10905, 10908, 11080, 11085, 11088, 11170, 11172, 11175, 11178, 11260, 11265, 11268, 11350, 11352, 11354, 11355, 11358, 11440, 11445, 11448, 11530, 11532, 11535, 11536, 11538, 11620, 11625, 11628, 11710, 11712, 11713, 11715, 11718, 11800, 11805, 11808, 12070, 12072, 12075, 12078, 12160, 12165, 12168, 12250, 12252, 12255, 12256, 12257, 12258, 12340, 12345, 12348, 12430, 12432, 12435, 12438, 12520, 12525, 12528, 12610, 12612, 12614, 12615, 12618, 12700, 12705, 12708, 13060, 13065, 13068, 13150, 13152, 13155, 13158, 13240, 13244, 13245, 13248, 13330, 13332, 13335, 13338, 13420, 13425, 13428, 13510, 13512, 13515, 13518, 13600, 13605, 13608, 14050, 14052, 14053, 14055, 14058, 14140, 14145, 14148, 14230, 14232, 14235, 14238, 14320, 14325, 14328, 14410, 14412, 14415, 14416, 14418, 14500, 14504, 14505, 14508, 15040, 15045, 15048, 15130, 15132, 15134, 15135, 15136, 15138, 15220, 15223, 15225, 15228, 15310, 15312, 15315, 15317, 15318, 15400, 15405, 15408, 15409, 16030, 16032, 16035, 16038, 16120, 16125, 16128, 16210, 16212, 16215, 16218, 16300, 16305, 16308, 17020, 17024, 17025, 17028, 17110, 17112, 17115, 17118, 17119, 17200, 17205, 17208, 18010, 18012, 18015, 18016, 18018, 18100, 18105, 18108, 19000, 19005, 19008, 21897, 21980, 21984, 22792, 22793, 22880, 22975, 22977, 23780, 23782, 23784, 23787, 23875, 23960, 23961, 23968, 24591, 24596, 24597, 24680, 24772, 24775, 24860, 24863, 24864, 24867, 25580, 25584, 25675, 25677, 25760, 25762, 25766, 25851, 25940, 25944, 25947, 26480, 26481, 26487, 26488, 26575, 26660, 26664, 26752, 26757, 26840, 26933, 26936, 27297, 27380, 27384, 27475, 27560, 27567, 27740, 27741, 27742, 27744, 27748, 27837, 27920, 27925, 28280, 28371, 28375, 28377, 28460, 28464, 28640, 28647, 28732, 28739, 28820, 28824, 28825, 28917, 29180, 29184, 29187, 29275, 29276, 29360, 29457, 29540, 29544, 29631, 29720, 29722, 29725, 29727, 29900, 29904, 39990, 39996, 39997

For example, 39997 has a first and last digit concatenation of 37 with a digit of 37 and a factoriation of 23 x 37 x 47. 

If instead, we require the product of the number's digit instead then only five numbers satisfy: 135, 11232, 12132, 12312, 13212 (permalink). For example, 13212 = 2^2 x 3^2 x 367 has a digit product of 12 which is a divisor of the number and 12 is also the concatenation of the first and last digits of the number.

Another criterion that could be applied is to require that the first and last digits be the same. In this case 227 numbers satisfy in the range up to 40000. They are (permalink):

121, 242, 363, 484, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 3003, 3333, 3663, 3993, 4004, 4224, 4444, 4664, 4884, 5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995, 6006, 6336, 6666, 6996, 7007, 7777, 8008, 8448, 8888, 9009, 9999, 10021, 10131, 10241, 10351, 10461, 10571, 10681, 10791, 10901, 11011, 11121, 11231, 11341, 11451, 11561, 11671, 11781, 11891, 12001, 12111, 12221, 12331, 12441, 12551, 12661, 12771, 12881, 12991, 13101, 13211, 13321, 13431, 13541, 13651, 13761, 13871, 13981, 14091, 14201, 14311, 14421, 14531, 14641, 14751, 14861, 14971, 15081, 15191, 15301, 15411, 15521, 15631, 15741, 15851, 15961, 16071, 16181, 16291, 16401, 16511, 16621, 16731, 16841, 16951, 17061, 17171, 17281, 17391, 17501, 17611, 17721, 17831, 17941, 18051, 18161, 18271, 18381, 18491, 18601, 18711, 18821, 18931, 19041, 19151, 19261, 19371, 19481, 19591, 19701, 19811, 19921, 20042, 20152, 20262, 20372, 20482, 20592, 20702, 20812, 20922, 21032, 21142, 21252, 21362, 21472, 21582, 21692, 21802, 21912, 22022, 22132, 22242, 22352, 22462, 22572, 22682, 22792, 22902, 23012, 23122, 23232, 23342, 23452, 23562, 23672, 23782, 23892, 24002, 24112, 24222, 24332, 24442, 24552, 24662, 24772, 24882, 24992, 25102, 25212, 25322, 25432, 25542, 25652, 25762, 25872, 25982, 26092, 26202, 26312, 26422, 26532, 26642, 26752, 26862, 26972, 27082, 27192, 27302, 27412, 27522, 27632, 27742, 27852, 27962, 28072, 28182, 28292, 28402, 28512, 28622, 28732, 28842, 28952, 29062, 29172, 29282, 29392, 29502, 29612, 29722, 29832, 29942, 30063, 30393, 30723, 31053, 31383, 31713, 32043, 32373, 32703, 33033, 33363, 33693, 34023, 34353, 34683, 35013, 35343, 35673, 36003, 36333, 36663, 36993, 37323, 37653, 37983, 38313, 38643, 38973, 39303, 39633, 39963

For example, 39963 = 3 x 7 x 11 x 173 has first and last digits the same and 33 divides the number.

Jordan-Polya Numbers

The number associated with my diurnal age today, 27648, is a Jordan-Polya number which Numbers Aplenty defines as a number that can be written as the product of factorial numbers. In the case of 27648 we have:$$\begin{align} 27648 &=2^{10} \times 3^3\\&=(4!)^3 \times 2! \end{align} $$There aren't that many of them in the range up to 40,000. Here are the initial numbers:

1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, 288, 384, 432, 480, 512, 576, 720, 768, 864, 960, 1024, 1152, 1296, 1440, 1536, 1728, 1920, 2048, 2304, 2592, 2880, 3072, 3456, 3840, 4096, 4320, 4608, 5040, 5184, 5760, 6144, 6912, 7680, 7776, 8192, 8640, 9216, 10080, 10368, 11520, 12288, 13824, 14400, 15360, 15552, 16384, 17280, 18432, 20160, 20736, 23040, 24576, 25920, 27648, 28800, 30240, 30720, 31104, 32768, 34560, 36864

Here is what Wikipedia had to say about them:

In mathematics, the Jordan–Pólya numbers are the numbers that can be obtained by multiplying together one or more factorials, not required to be distinct from each other. For instance, 480 is a Jordan–Pólya number because:$$480 =2! \times 2! \times 5!$$Every tree has a number of symmetries that is a Jordan–Pólya number, and every Jordan–Pólya number arises in this way as the order of an automorphism group of a tree. These numbers are named after Camille Jordan and George Pólya, who both wrote about them in the context of symmetries of trees.

These numbers grow more quickly than polynomials but more slowly than exponentials. As well as in the symmetries of trees, they arise as the numbers of transitive orientations of comparability graphs and in the problem of finding factorials that can be represented as products of smaller factorials.

A tree that (as an abstract graph) has 480 symmetries (automorphisms). There are 2 ways of permuting the two children of the upper left vertex, 2 ways of permuting the two children of the upper middle vertex, and 5! = 120 ways of permuting the five children of the upper right vertex, for 2 x 2 x 120 = 480 symmetries altogether.

George Pólya
13 December 1887 - 7 September 1985

George Pólya was one of the greatest mathematicians of the 20th century and there is a brief biography of his life to be found here. To quote from this article:

With no hesitation, George Pólya is my personal hero as a mathematician. ... [he] is not only a distinguished gentleman but a most kind and gentle man: his ebullient enthusiasm, the twinkle in his eye, his tremendous curiosity, his generosity with his time, his spry energetic walk, his warm genuine friendliness, his welcoming visitors into his home and showing them his pictures of great mathematicians he has known - these are all components of his happy personality. As a mathematician, his depth, speed, brilliance, versatility, power and universality are all inspiring. Would that there were a way of teaching and learning these traits.

Camille Jordan
5th January 1838 - 22nd January 1922

Camille Jordan made most of his mathematical contributions in the 19th century and a brief biography can be found here. It can be noted that, quoting from the article:

The Journal de Mathématiques Pure et Appliquées was a leading mathematical journal and played a very significant part in the development of mathematics throughout the 19th century. It was usually known as the Journal de Liouville since Liouville had founded the journal in 1836. Liouville died in 1882 and in 1885 Jordan became editor of the Journal, a role he kept for over 35 years until his death.

Prime Sums Of Digits, Digits Squared And Digits Cubed

Consider the number 27640 (my diurnal age today) with the property that the sum of its digits cubed is a prime number:$$ \begin{align} 27460 &= 2^3+7^3+6^3+4^3 +0^3\\ &= 631 \text{ (prime)} \end{align} $$However, its sum of digits squared is composite:$$ \begin{align} 27460 &= 2^2+7^2+6^3+4^2 +0^2\\ &= 105 \\&=105 \\ &= 3 \times 5 \times 7 \end{align}$$Then again, its sum of digits is prime:$$ \begin{align} 27460 &= 2+7+6+4 +0\\ &= 19 \text{ (prime)} \end{align} $$So the question can be asked as to what numbers, in the range up to 40,000, have the property that the:

• sum of digits is a prime number
• sum of digits squared is a prime number
• sum of digits cubed is a prime number

It turns out that there are 1985 such numbers and they belong to OEIS A245475:

A245475: numbers \(n\) such that the sum of digits, sum of squares of digits, and sum of cubes of digits are all prime.

The sequence begins (permalink):

11, 101, 110, 111, 113, 131, 146, 164, 166, 199, 223, 232, 289, 298, 311, 322, 335, 337, 346, 353, 355, 364, 373, 388, 416, 436, 449, 461, 463, 494, 533, 535, 553, 566, 614, 616, 634, 641, 643, 656, 661, 665, 733, 829, 838, 883, 892, 919, 928, 944, 982, 991, 1001, 1010, 1011, 1013, 1031, 1046, 1064, 1066, 1099

Of these 1985 numbers in the range up to 40,000, 322 are prime themselves. However, many of these numbers are permutations of the digits of other numbers and so it's pertinent to ask how many "root numbers" are there in the range up to 40,000. These "root numbers" will be numbers whose digits are in ascending order such that any permutation of their digits still ensures membership in OEIS A245475. There are only 99 such numbers and they are:

11, 111, 113, 146, 166, 199, 223, 289, 335, 337, 346, 355, 388, 449, 566, 1112, 1114, 1145, 1147, 1244, 1349, 1448, 1499, 1679, 2225, 2227, 2258, 2333, 2458, 2555, 2557, 3347, 3358, 3367, 3466, 3569, 4445, 4667, 5558, 7888, 11111, 11113, 11117, 11122, 11126, 11128, 11137, 11159, 11234, 11245, 11333, 11344, 11366, 11399, 11489, 11777, 12224, 12248, 12347, 12358, 12446, 12488, 12677, 13333, 13339, 13445, 13478, 14455, 14459, 14558, 16888, 17777, 17779, 22225, 22229, 22258, 22348, 22355, 22447, 22577, 22999, 23444, 23455, 23558, 23699, 24779, 24788, 25589, 25688, 28999, 33335, 33368, 33449, 33689, 34444, 34466, 35777, 36668, 36679

Note that these numbers do not contain any zeroes. This is because we can insert any number of zeroes anywhere we want. For example:$$11 \to 10010$$Of these, the following have digits that are all different:$$146, 289, 346, 1349, 1679, 2458, 3569, 12347, 12358, 13478$$

Concatenated Triples Of Cubes and Doubles of Squares

The number associated with my diurnal age today, 27640, has the property that it is a sum of concatentated cubes:$$ \begin{align} 27640 &= 27 \, | \,64  \,|\, 0\\ &= 3^3  \,| \, 4^3 \, |\, 0^3 \end{align}$$I then sought to discover what other numbers in the range up to 40,000 have this property. It turns out that there are 78 such numbers and they are (permalink):

108, 180, 801, 810, 1027, 1064, 1270, 1278, 1640, 1648, 1827, 1864, 2701, 2708, 2710, 2718, 2780, 2781, 6401, 6408, 6410, 6418, 6480, 6481, 8027, 8064, 8127, 8164, 8270, 8271, 8640, 8641, 10125, 10216, 10343, 10512, 10729, 11250, 11258, 12160, 12168, 12501, 12508, 12510, 12518, 12580, 12581, 12764, 13430, 13438, 15120, 15128, 16427, 17290, 17298, 18125, 18216, 18343, 18512, 18729, 21601, 21608, 21610, 21618, 21680, 21681, 27064, 27164, 27640, 27641, 27648, 27864, 34301, 34308, 34310, 34318, 34380, 34381

Let's take the last member of the above set:$$ \begin{align} 34381 &= 343 \, | \, 8 \, | \, 1\\ & =7^3 \, | \, 2^3 \, | \, 1^3 \end{align} $$It can be noted that 27640 and 27641 form a pair of consecutive numbers sharing this same property. In the range up to 40,000 this pairing occurs as shown:

• 2780 and 2781
• 6480 and 6481
• 8270 and 8271
• 8640 and 8641
• 12580 and 12581
• 21680 and 21681
• 27640 and 27641
• 34380 and 34381

If instead of cubes, we consider squares and instead of triples we consider doubles then we have the following 557 numbers in the range up to 40,000 (permalink):

10, 14, 19, 40, 41, 49, 90, 91, 94, 116, 125, 136, 149, 160, 161, 164, 164, 169, 181, 250, 251, 254, 259, 360, 361, 364, 369, 416, 425, 436, 449, 464, 481, 490, 491, 494, 499, 640, 641, 644, 649, 810, 811, 814, 819, 916, 925, 936, 949, 964, 981, 1000, 1001, 1004, 1009, 1100, 1121, 1144, 1169, 1196, 1210, 1211, 1214, 1219, 1225, 1256, 1289, 1324, 1361, 1400, 1440, 1441, 1441, 1444, 1449, 1484, 1529, 1576, 1625, 1625, 1636, 1649, 1664, 1676, 1681, 1690, 1691, 1694, 1699, 1729, 1784, 1841, 1900, 1960, 1961, 1961, 1964, 1969, 2250, 2251, 2254, 2259, 2516, 2536, 2549, 2560, 2561, 2564, 2564, 2569, 2581, 2890, 2891, 2894, 2899, 3240, 3241, 3244, 3249, 3610, 3611, 3614, 3616, 3619, 3625, 3649, 3664, 3681, 4000, 4001, 4004, 4009, 4100, 4121, 4144, 4169, 4196, 4225, 4256, 4289, 4324, 4361, 4400, 4410, 4411, 4414, 4419, 4441, 4484, 4529, 4576, 4625, 4676, 4729, 4784, 4840, 4841, 4841, 4844, 4849, 4900, 4916, 4925, 4936, 4961, 4964, 4981, 5290, 5291, 5294, 5299, 5760, 5761, 5764, 5769, 6250, 6251, 6254, 6259, 6416, 6425, 6436, 6449, 6481, 6760, 6761, 6764, 6769, 7290, 7291, 7294, 7299, 7840, 7841, 7844, 7849, 8116, 8125, 8136, 8149, 8164, 8410, 8411, 8414, 8419, 9000, 9001, 9004, 9009, 9100, 9121, 9144, 9169, 9196, 9225, 9256, 9289, 9324, 9361, 9400, 9441, 9484, 9529, 9576, 9610, 9611, 9614, 9619, 9625, 9676, 9729, 9784, 9841, 9900, 9961, 10016, 10025, 10036, 10049, 10064, 10081, 10240, 10241, 10244, 10249, 10890, 10891, 10894, 10899, 11024, 11089, 11156, 11225, 11296, 11369, 11444, 11521, 11560, 11561, 11564, 11569, 11600, 11681, 11764, 11849, 11936, 12025, 12116, 12116, 12125, 12136, 12149, 12164, 12181, 12209, 12250, 12251, 12254, 12259, 12304, 12401, 12500, 12601, 12704, 12809, 12916, 12960, 12961, 12964, 12969, 13025, 13136, 13249, 13364, 13481, 13600, 13690, 13691, 13694, 13699, 13721, 13844, 13969, 14096, 14225, 14356, 14416, 14425, 14436, 14440, 14441, 14444, 14449, 14449, 14464, 14481, 14489, 14624, 14761, 14900, 15041, 15184, 15210, 15211, 15214, 15219, 15329, 15476, 15625, 15776, 15929, 16000, 16001, 16004, 16009, 16084, 16100, 16121, 16144, 16169, 16196, 16225, 16241, 16256, 16289, 16324, 16361, 16400, 16400, 16441, 16484, 16529, 16561, 16576, 16625, 16676, 16724, 16729, 16784, 16810, 16811, 16814, 16819, 16841, 16889, 16900, 16916, 16925, 16936, 16949, 16961, 16964, 16981, 17056, 17225, 17396, 17569, 17640, 17641, 17644, 17649, 17744, 17921, 18100, 18281, 18464, 18490, 18491, 18494, 18499, 18649, 18836, 19025, 19216, 19360, 19361, 19364, 19369, 19409, 19604, 19616, 19625, 19636, 19649, 19664, 19681, 19801, 20250, 20251, 20254, 20259, 21160, 21161, 21164, 21169, 22090, 22091, 22094, 22099, 22516, 22525, 22536, 22549, 22564, 22581, 23040, 23041, 23044, 23049, 24010, 24011, 24014, 24019, 25000, 25001, 25004, 25009, 25100, 25121, 25144, 25169, 25196, 25225, 25256, 25289, 25324, 25361, 25400, 25441, 25484, 25529, 25576, 25616, 25625, 25625, 25636, 25649, 25664, 25676, 25681, 25729, 25784, 25841, 25900, 25961, 26010, 26011, 26014, 26019, 27040, 27041, 27044, 27049, 28090, 28091, 28094, 28099, 28916, 28925, 28936, 28949, 28964, 28981, 29160, 29161, 29164, 29169, 30250, 30251, 30254, 30259, 31360, 31361, 31364, 31369, 32416, 32425, 32436, 32449, 32464, 32481, 32490, 32491, 32494, 32499, 33640, 33641, 33644, 33649, 34810, 34811, 34814, 34819, 36000, 36001, 36004, 36009, 36100, 36116, 36121, 36125, 36136, 36144, 36149, 36164, 36169, 36181, 36196, 36225, 36256, 36289, 36324, 36361, 36400, 36441, 36484, 36529, 36576, 36625, 36676, 36729, 36784, 36841, 36900, 36961, 37210, 37211, 37214, 37219, 38440, 38441, 38444, 38449, 39690, 39691, 39694, 39699

Let's take the last number in the above set of numbers:$$ \begin{align} 39699 &= 3969 \, | \, 9 \\ &= 63^2 \, | \, 3^2 \end{align}$$One more example is:$$ \begin{align} 36000 &= 3600 \, | \, 0 \\ &= 60^2 \, | \, 0^2 \end{align} $$There are obviously many other variation on these themes but that will suffice for now.

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