Palindromic Day 28182

Palindromic properties of 28182 (showing only sequence members up to 40000):

A098834: palindromic Smith numbers.

4, 22, 121, 202, 454, 535, 636, 666, 1111, 1881, 3663, 7227, 7447, 9229, 10201, 17271, 22522, 24142, 28182, 33633, 38283

A Smith number is a composite number where the sum of its digits equals the sum of the digits of its prime factors. For 28182:$$ \begin{align} 28182 &\rightarrow 2+8+1+8+2 = 21 \\ 28182 &= 2 \times 3 \times 7 \times 11 \times 61\\ &\rightarrow 2 + 3 + 7 + 1+1+6 +1 =21 \end{align}$$

A046395: palindromes that are the product of 5 distinct primes.

6006, 8778, 20202, 28182

Here \(28182 = 2 \times 3 \times 7 \times 11 \times 61 \)

A099052: all palindromes of length > 1 in the decimal expansion of \(e\).

\(e\) = 2.71828182845904523536028747135266 ...

A045571: numbers that are palindromic, divisible by 11 and have an odd number of digits.

121, 242, 363, 484, 616, 737, 858, 979, 10901, 11011, 12221, 13431, 14641, 15851, 17171, 18381, 19591, 20702, 21912, 22022, 23232, 24442, 25652, 26862, 28182, 29392, 30503, 31713, 32923, 33033, 34243, 35453, 36663, 37873, 39193

All the palindromic numbers with an even number of digits are divisible by 11. The number of palindromic numbers with \(2k+1\) digits that are divisible by 11 is \((10^{k+1} + (-1)^k)/11\), and their asymptotic relative density within the set of all palindromic numbers with an odd number of digits is 1/11 (from OEIS comments).

A113838: palindromes sandwiched between twin primes.

4, 6, 282, 828, 858, 2112, 21012, 21612, 23832, 26262, 26862, 28182

Here of course the twin primes are 28181 and 28183.

A032751: palindromic Super-3 Numbers.

4554, 6776, 17471, 22322, 22722, 28182

Super-3 numbers \(n\) are of the form \(3 \times n^3 \) and contain three consecutive 3's.

Here \(3 \times 28182^3 = 67148557\textbf{333}704\)

Other Special Classes of Interprimes

On the 1st November 2023, I posted on A Special Class of Interprime and these were non-palindromic composite numbers located between twin primes which, when reversed, are also located between twin primes. Some work both ways while some are only one way because they end in a zero. Figure 1 shows an example of the former while Figure 2 shows an example of the latter.

Figure 1

Figure 2

Today I turned 27738 days old and this number is an interprime number between twin primes which when concatenated with itself forms a number which is also an interprime between twin primes. The result for 27738 is shown in Figure 3.

Figure 3

Numbers of this sort belong to OEIS A235109 :

A235109     Averages q of twin prime pairs, such that q concatenated to q is also the average of a twin prime pair.

The initial members are (permalink):

42, 102, 108, 180, 192, 270, 312, 420, 522, 660, 822, 882, 1230, 1482, 4242, 4788, 8820, 10332, 11550, 13692, 14550, 14562, 14868, 15732, 17910, 18522, 20550, 21648, 22620, 23670, 23832, 26262, 27738, 35838, 38922, 39042, 40128, 42018, 43962, 44532, 46440

As a variation on this, we could concatenate an interprime with its reversal, thus forming a palindrome. This is shown in Figure 4.

Figure 4

Up to 40000, the initial interprimes with this property are (permalink) 240, 270, 2142, 8388, 22092, 22962, 23832, 24420, 24918, 26262, 27690 and 28110. The sequence does not appear in the OEIS. The members of this sequence are, to be fair, rather sparse and could be made more numerous if the condition that the interprime lay between twin primes was relaxed. If we simply require that the interprime, when concatenated with its reverse, is also an interprime then in the range up to 40000, 263 numbers satisfy. The numbers are (permalink):

9, 15, 21, 42, 93, 102, 105, 108, 160, 240, 246, 270, 279, 324, 386, 432, 754, 810, 909, 933, 1092, 1302, 1452, 1611, 1998, 2142, 2205, 2295, 2322, 2336, 2470, 2568, 2667, 2892, 2900, 2946, 3021, 3326, 3423, 3453, 3465, 3558, 3588, 3627, 3672, 3736, 3885, 3921, 4002, 4065, 4076, 4131, 4353, 4422, 4646, 4742, 4785, 5193, 5439, 5481, 5502, 5529, 5607, 5804, 6107, 6340, 6376, 6798, 6969, 7182, 7212, 7494, 8097, 8169, 8388, 8437, 8844, 8908, 8985, 9394, 9678, 9865, 10008, 10101, 10794, 10815, 10875, 10944, 10998, 11226, 11445, 11523, 11817, 12024, 12111, 12252, 12489, 12500, 12514, 12826, 12947, 13056, 13101, 13320, 13374, 13482, 13560, 13674, 13740, 13881, 13965, 14064, 14415, 14592, 14715, 15015, 15087, 15534, 15664, 16230, 16396, 16799, 17388, 17529, 17958, 18042, 18288, 18360, 18447, 18531, 18737, 19149, 19314, 19548, 19704, 19857, 20022, 20049, 20057, 20225, 20358, 20403, 20687, 20745, 20751, 20808, 21015, 21104, 21189, 21202, 21381, 21404, 21558, 21969, 22092, 22272, 22719, 22866, 22904, 22962, 23124, 23631, 23832, 24036, 24144, 24333, 24420, 24522, 24804, 24855, 24918, 25001, 25080, 25305, 25417, 25455, 25470, 25578, 25595, 25761, 25932, 25960, 26180, 26262, 26412, 26582, 26637, 26675, 26748, 27075, 27429, 27546, 27597, 27690, 27999, 28110, 28117, 28253, 28314, 28410, 28629, 28692, 28869, 29247, 29577, 29720, 29826, 29865, 29937, 30106, 30165, 30217, 30270, 30693, 31149, 31152, 31182, 31269, 31536, 31617, 31653, 31977, 32244, 32325, 32700, 32914, 33186, 33288, 33573, 33588, 33621, 33639, 33854, 33939, 34125, 34290, 34412, 34590, 34683, 34743, 34874, 34962, 35094, 35421, 35674, 35802, 36003, 36442, 36486, 36648, 36694, 36764, 37220, 37514, 37548, 38385, 38856, 39093, 39159, 39447, 39627, 39852, 39999

Let's take 93 from the previous list as an example. It is an interprime that lies midway between 89 and 97. Concatenated with its reverse (39), we get 9339 and this number is midway between 9337 and 9341. Figure 5 illustrates this.

Figure 5

Similarly we could relax the interprime condition for interprimes that are concatenated with themselves (but not reversed). There are 345 interprimes in the range up to 40000 that qualify. An example is 21, an interprime between 19 and 23, that forms 2121, an interprime between 2113 and 2129.

Figure 6

The interprimes between 27700 and 40000 with this property are (permalink):

..., 27738, 27888, 27945, 27990, 28281, 28515, 28613, 28740, 28815, 28851, 28994, 29013, 29170, 29237, 29307, 29448, 29835, 29953, 30000, 30038, 30264, 30300, 30310, 30378, 30468, 30555, 30846, 30856, 30902, 31080, 31122, 31269, 31335, 31347, 31660, 31854, 31960, 32298, 32361, 32715, 32925, 32990, 33018, 33235, 33351, 33465, 33594, 33717, 33840, 33860, 34224, 34734, 34743, 34848, 35325, 35556, 35571, 35838, 35980, 36189, 36462, 36680, 37008, 37053, 37176, 37576, 37850, 38076, 38238, 38310, 38331, 38685, 38922, 39042, 39084, 39093, 39105, 39363, 39378, 39447, 39546, 39691, 39765, 39774, 39894, ...

Lastly, if we relax the interprime condition that the interprime and its reverse must lie between twin primes, then there are 629 numbers that satisfy in the range up to 40000 (permalink) but I won't list those here.

A Special Number Plate

Last night I noticed an unusual car number plate. It was 432 432.  This is a customised number plates as standard number plates follow an AAA 000 pattern, that is three uppercase letters followed by three digits. Presumably the number 432 was of some significance to the person who purchased the plates. This got me thinking about what is special about the number 432.

432 432

PROPERTY 1

The first property of interest is that it's wedged between two prime numbers: 431 and 433. Thus we have:$$432 = \frac{431+433}{2}$$This qualifies it for membership in OEIS A014574:

A014574

Average of twin prime pairs.

The initial members of this sequence are:

4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, 312, 348, 420, 432, 462, 522, 570, 600, 618, 642, 660, 810, 822, 828, 858, 882, 1020, 1032, 1050, 1062, 1092, 1152, 1230, 1278, 1290, 1302, 1320, 1428, 1452, 1482, 1488, 1608

PROPERTY 2

The next property of interest is that it's the sum of two cubes. Specifically:$$432=6^3+6^3$$This property qualifies it for membership in OEIS A003325:

A003325

Numbers that are the sum of 2 positive cubes.

The initial members of this sequence are:

2, 9, 16, 28, 35, 54, 65, 72, 91, 126, 128, 133, 152, 189, 217, 224, 243, 250, 280, 341, 344, 351, 370, 407, 432, 468, 513, 520, 539, 559, 576, 637, 686, 728, 730, 737, 756, 793, 854, 855, 945, 1001, 1008, 1024, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1343

PROPERTY 3

The next property is that it's the sum of the totients of the first 37 numbers: $$432=\sum_{n=1}^{37} \phi(n)$$This qualifies it for membership in OEIS A002088:

A002088

Sum of totient function: \( \displaystyle{\text{a}(n) = \sum_{k=1}^n \phi(k) } \)

The initial members of the sequence are:

0, 1, 2, 4, 6, 10, 12, 18, 22, 28, 32, 42, 46, 58, 64, 72, 80, 96, 102, 120, 128, 140, 150, 172, 180, 200, 212, 230, 242, 270, 278, 308, 324, 344, 360, 384, 396, 432, 450, 474, 490, 530, 542, 584, 604, 628, 650, 696, 712, 754, 774, 806, 830, 882, 900, 940, 964

PROPERTY 4

  

Another property makes it a member of OEIS A033833:

A033833

Highly factorizable numbers: numbers with a record number of proper factorizations.

432 turns out to have 56 possible factors which are:

[2, 2, 2, 2, 3, 3, 3], [2, 2, 2, 2, 3, 9], [2, 2, 2, 2, 27], [2, 2, 2, 3, 3, 6], [2, 2, 2, 3, 18], [2, 2, 2, 6, 9], [2, 2 , 2, 54], [2, 2, 3, 3, 3, 4], [2, 2, 3, 3, 12], [2, 2, 3, 4, 9], [2, 2, 3, 6, 6], [2, 2, 3, 36], [2, 2, 4, 27], [2, 2, 6, 18], [2, 2, 9, 12], [2, 2, 108], [2, 3, 3, 3, 8], [2, 3, 3, 4, 6], [2, 3, 3, 24], [2, 3, 4, 18], [2, 3, 6, 12], [2, 3, 8, 9], [2, 3, 72], [ 2, 4, 6, 9], [2, 4, 54], [2, 6, 6, 6], [2, 6, 36], [2, 8, 27], [2, 9, 4], [2, 12, 18], [2, 216], [3, 3, 3, 4, 4], [3, 3, 3, 16], [3, 3, 4, 12], [3, 3, 6, 8], [3, 3, 48], [3, 4, 4, 9], [3, 4, 6, 6], [3, 4, 36], [3, 6, 24], [3, 8, 18], [3, 9, 16], [3, 12, 12], [3, 144], [4, 4, 27], [4, 6, 18], [4, 9, 12], [4, 108], [6, 6, 12], [6, 8, 9], [6, 72], [8, 54], [9, 48], [12, 36], [16, 27], [18, 24]

It can be noted that 666 makes its appearance since 432 = 2 x 6 x 6 x 6.

The initial members of the sequence are:

1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 192, 216, 240, 288, 360, 432, 480, 576, 720, 960, 1080, 1152, 1440, 2160, 2880, 4320, 5040, 5760, 7200, 8640, 10080, 11520, 12960, 14400, 15120, 17280, 20160, 25920, 28800, 30240, 34560

PROPERTY 5

Another property of 432 is that it's the difference between the squares of two successive primes. Specifically$$ \begin{align} 432&=109^2-107^2\\&=(109+107) \times (109-107)\\&=216 \times 2\\ &=2^4 \times 3^3 \end{align} $$This qualifies 432 for inclusion in OEIS A069482:

A069482

a(\(n\)) = (prime(\(n\)+1))\(^2\) - (prime(\(n\)))\(^2\)

The initial members of the sequence are:

5, 16, 24, 72, 48, 120, 72, 168, 312, 120, 408, 312, 168, 360, 600, 672, 240, 768, 552, 288, 912, 648, 1032, 1488, 792, 408, 840, 432, 888, 3360, 1032, 1608, 552, 2880, 600, 1848, 1920, 1320, 2040, 2112, 720, 3720, 768, 1560, 792, 4920, 5208, 1800, 912, 1848

I'll stop there as I think I've shown that 432 has at least five interesting properties but of course there are many more. There are actually 4018 entries for this number in the OEIS.

SOD Prime Chains

Over the years, I've looked at many forms of prime chains but, as far as I know, not prime chains formed by successively adding the sum of the digits of the prime. What got me thinking about this type of prime chain was the number associated with my diurnal age today: 27043. This number is prime and if we add its sum of digits, we get a new prime. Thus, where SOD stands for Sum Of Digits, we have:$$ \overbrace{27043}^{\text{prime}} + \overbrace{16}^{\text{SOD}}=27059 \text{ which is also prime}$$Because 27059 is the next prime after 27043, it is known as an \( \textbf{a-pointer prime} \) defined by Numbers Aplenty as follows:

A prime number  \(p\) is called a-pointer if the next prime number can be obtained adding  \(p\)  to its sum of digits (here the 'a' stands for additive).

When considering prime chains formed by adding the sum of digits, we are only interested in "prime-ness" and not "a-pointer prime-ness". The earliest example of a prime chain begins with the prime 11. If we add its sum of digits, we get 13 and thus we have a prime chain of length 1: $$\overbrace{11}^{\text{prime}}+\overbrace{2}^{\text{SOD}}=\overbrace{13}^{\text{prime}}$$If we add the sum of digits again we get 17 and thus we have a chain of length 2 namely: $$\overbrace{11}^{\text{prime}}+\overbrace{2}^{\text{SOD}}=\overbrace{13}^{\text{prime}} \text{ and } \overbrace{13}^{\text{prime}}+\overbrace{4}^{\text{SOD}}=\overbrace{17}^{\text{prime}}$$Here, both 11 and 13 are a-pointer primes. We do not get a chain of three primes until 277 where the chain is:$$ 277 \rightarrow 293 \rightarrow 307 \rightarrow 317$$None of these primes are a-pointer primes. The first chain of four occurs with 37783:$$37783 \rightarrow 37811 \rightarrow 37831 \rightarrow 37853 \rightarrow 37879$$The first chain of five occurs with 516493:$$516493 \rightarrow 516521 \rightarrow 516541 \rightarrow 516563 \rightarrow 516589 \rightarrow 516623$$These record chains constitute OEIS A090009:

A090009

Begins the earliest length-\(n\) chain of primes such that any term in the chain equals the previous term increased by the sum of its digits.

The initial members are (permalink - will time out beyond 516493):

2, 11, 11, 277, 37783, 516493, 286330897, 286330897, 56676324799

The progressions for the larger numbers are:

• 286330897 286330943 286330981 286331021 286331047 286331081 286331113 286331141 
• 56676324799 56676324863 56676324919 56676324977 56676325039 56676325091 56676325141 56676325187 56676325243 

In conclusion, we must say that 27043 has the unusual property that its successor, 27044, also produces a prime (27061) when its sum of digits is added. Thus 27059 and 27061 form a pair of twin primes. Given this property of 27043, a new sequence could be formulated as follows:

Numbers \(n\) such that \(n\) plus digit sum of \(n\) and \(n+1\) plus digit sum of \(n+1\) are both prime.

These numbers constitute about 1.335% of numbers in the range up to 40000. This is to be expected since the probability of any number having this property is about 0.1, so two in succession would have a probability of about 0.01. The primes resulting from this process are generally twin primes, although perhaps not exclusively. Numbers ending in 9 such as 299 (with sod = 20) will change to 300 (with sod = 3). However, looking at the output below, there are no numbers ending in 9. Interesting. Triplets are not possible as this would mean three successive primes separated by only a single number. The 534 members up to 40000 are:

10, 13, 34, 52, 58, 91, 94, 100, 103, 127, 142, 166, 181, 184, 217, 232, 256, 271, 295, 304, 340, 412, 418, 451, 508, 583, 610, 631, 787, 811, 814, 838, 1024, 1042, 1048, 1081, 1138, 1222, 1264, 1285, 1312, 1420, 1441, 1465, 1468, 1591, 1597, 1600, 1606, 1648, 1681, 1711, 1771, 1861, 1915, 1933, 1975, 2017, 2071, 2074, 2095, 2104, 2122, 2128, 2230, 2254, 2293, 2302, 2326, 2365, 2638, 2671, 2692, 2701, 2767, 2782, 2947, 2980, 3112, 3154, 3241, 3244, 3283, 3316, 3355, 3373, 3445, 3448, 3514, 3538, 3751, 3796, 3805, 3913, 3976, 3994, 4012, 4036, 4075, 4120, 4144, 4210, 4216, 4231, 4255, 4324, 4411, 4495, 4504, 4528, 4618, 4633, 4696, 4705, 4765, 4780, 4945, 4984, 5002, 5008, 5083, 5221, 5263, 5395, 5404, 5425, 5461, 5482, 5623, 5641, 5827, 5845, 5860, 6073, 6121, 6181, 6253, 6277, 6343, 6433, 6547, 6637, 6670, 6676, 6742, 6760, 6766, 6811, 6850, 6925, 6940, 7114, 7192, 7201, 7285, 7315, 7333, 7441, 7465, 7531, 7537, 7570, 7735, 7930, 7978, 8071, 8215, 8272, 8365, 8413, 8521, 8611, 8788, 8815, 8836, 8944, 8968, 8983, 9001, 9025, 9223, 9262, 9394, 9403, 9421, 9442, 9598, 9607, 9688, 9799, 9910, 9976, 10003, 10027, 10060, 10081, 10132, 10261, 10285, 10318, 10420, 10444, 10483, 10516, 10687, 10843, 10867, 10918, 11050, 11056, 11098, 11107, 11146, 11161, 11341, 11473, 11677, 11695, 11704, 11761, 11815, 11923, 11947, 12028, 12061, 12088, 12151, 12226, 12241, 12358, 12592, 12601, 12796, 12805, 12976, 13210, 13324, 13381, 13657, 13672, 13690, 13696, 13705, 13741, 13813, 13855, 13876, 13984, 14002, 14065, 14311, 14371, 14428, 14533, 14572, 14608, 14845, 15124, 15253, 15271, 15343, 15499, 15562, 15631, 15721, 15946, 16045, 16048, 16171, 16399, 16627, 16666, 16798, 16807, 16954, 16996, 17014, 17167, 17185, 17272, 17365, 17470, 17560, 17635, 17656, 17725, 17764, 17815, 17878, 17893, 17902, 17962, 18025, 18028, 18043, 18115, 18265, 18286, 18517, 18883, 18886, 19057, 19123, 19162, 19186, 19360, 19411, 19450, 19522, 19672, 19726, 19816, 19858, 19963, 19987, 20014, 20140, 20218, 20344, 20431, 20458, 20491, 20500, 20695, 20704, 20728, 20785, 20875, 20962, 20986, 21004, 21007, 21046, 21175, 21310, 21358, 21511, 21538, 21571, 21577, 21592, 21601, 21628, 21718, 21823, 22030, 22075, 22093, 22102, 22144, 22255, 22258, 22348, 22525, 22618, 22675, 22723, 22837, 22942, 23020, 23026, 23044, 23353, 23518, 23608, 23647, 23665, 23884, 24091, 24100, 24163, 24892, 24901, 25021, 25153, 25282, 25285, 25390, 25447, 25555, 25576, 25774, 25825, 25912, 25972, 26092, 26101, 26233, 26656, 26674, 26698, 26707, 26836, 26854, 26926, 27043, 27085, 27223, 27262, 27460, 27511, 27517, 27556, 27664, 27715, 27730, 27886, 27916, 28075, 28090, 28162, 28255, 28327, 28384, 28525, 28546, 28588, 28636, 28726, 28987, 29005, 29113, 29374, 29644, 29734, 29848, 29977, 30004, 30130, 30373, 30448, 30538, 30823, 30847, 31108, 31141, 31165, 31234, 31297, 31306, 31492, 31501, 31525, 31696, 31705, 31708, 31747, 31831, 32020, 32044, 32110, 32131, 32173, 32281, 32311, 32353, 32392, 32401, 32425, 32515, 32590, 32776, 32884, 32917, 32950, 33055, 33163, 33271, 33328, 33565, 33580, 33727, 33745, 33784, 33811, 34021, 34114, 34138, 34192, 34201, 34243, 34282, 34351, 34447, 34480, 34486, 34570, 34627, 34735, 34822, 34825, 34936, 35035, 35257, 35299, 35431, 35566, 35698, 35707, 35815, 35872, 35983, 36001, 36085, 36445, 36511, 36754, 36868, 36910, 36991, 37180, 37321, 37342, 37525, 37546, 37564, 37783, 37963, 38221, 38311, 38425, 38440, 38578, 38626, 38644, 38683, 38887, 39142, 39211, 39217, 39322, 39346, 39478, 39814 

If we impose the restriction that \(n\) must be a prime number, then only 66 numbers qualify. Permalink. These numbers are:

13, 103, 127, 181, 271, 631, 787, 811, 1597, 1861, 1933, 2017, 2293, 2671, 2767, 3373, 4231, 5623, 5641, 5827, 6073, 6121, 6277, 6343, 6547, 6637, 7333, 7537, 8521, 9001, 9403, 9421, 10687, 10867, 11161, 11677, 11923, 12241, 12601, 13381, 14533, 15271, 17167, 18043, 18517, 19963, 20431, 21577, 21601, 22093, 24091, 25153, 25447, 27043, 32173, 32353, 32401, 32917, 33811, 34351, 35257, 35983, 37321, 37783, 37963, 39217

If we impose the restriction that \(n+1\) must be a prime number, then 73 numbers qualify. Permalink. These numbers are:

10, 52, 58, 100, 166, 232, 256, 418, 508, 838, 1048, 1222, 1600, 1606, 2128, 2692, 3448, 3538, 3796, 4012, 4210, 4216, 5002, 5008, 5482, 5860, 6760, 7192, 8272, 8836, 8968, 9688, 10060, 10132, 11056, 12226, 13690, 13696, 13876, 17470, 17656, 17902, 18286, 19162, 19726, 20218, 20962, 22030, 22258, 22348, 22618, 22942, 23020, 23026, 23608, 25390, 25576, 25912, 26698, 26926, 27916, 28162, 28546, 30448, 30538, 31306, 33328, 33580, 34282, 34486, 37180, 37546, 39322

Up to 10 million, no two consecutive prime numbers (that is a pair of twin primes) can produce another pair of twin primes.

Reversible Twin Prime Composites

What do I mean by a twin prime composite? Well, every twin prime pair has a composite number in between and that's the composite number I'm referring to. For example, 41 and 43 form a twin prime pair and 42 is the composite number in between. What's not so common though is a reversible twin prime composite meaning that when the composite is reversed, it's still wedged between two adjacent primes.

Now 42 fails in this regard because its reversal is 24 and while 23 is prime, 25 is not. However, 60 qualifies because its wedged between the primes 59 and 61 and, when it's reversed to form 6, it is still between two primes (5 and 7). Such numbers form OEIS sequence A103741:

A103741

Non-palindromic composites located between twin primes whose reverses, which are smaller, are also located between twin primes.

Here the condition that the reversed composite must be smaller than the original is imposed. The sequence up to one million is:

60, 240, 270, 600, 810, 822, 2130, 2340, 2802, 8010, 8220, 8430, 8838, 8862, 20550, 22740, 23202, 23370, 23910, 25410, 26880, 27240, 28410, 28572, 28662, 29022, 29760, 80472, 81702, 81930, 81972, 82140, 82530, 83220, 83340, 83640, 85620, 87222, 88470, 203430, 203460, 207240, 208590, 213360, 217200, 218970, 220020, 221070, 224910, 226902, 228300, 230940, 232080, 233160, 233550, 233940, 235440, 238080, 241260, 241512, 243432, 243702, 245130, 245910, 246510, 250050, 250950, 251970, 253680, 255180, 256722, 259122, 259620, 262050, 262152, 263610, 265542, 267390, 267960, 269220, 269430, 270240, 272010, 272760, 278562, 279552, 280410, 281250, 282240, 282390, 286542, 289020, 289140, 292710, 295200, 296730, 296970, 801000, 801420, 802650, 803730, 804282, 806370, 806382, 806790, 808020, 812760, 814062, 815412, 818580, 819618, 820320, 820680, 821208, 822762, 823830, 827130, 829728, 831540, 833712, 834150, 835320, 838092, 840180, 841020, 841080, 846060, 846360, 848790, 848922, 849600, 853902, 856548, 860010, 861492, 862482, 862650, 864300, 864630, 865638, 872658, 872748, 875340, 875418, 875520, 875760, 877110, 878022, 878832, 879168, 879582, 880068, 880248, 880800, 883410, 885552, 887400, 887658, 888060, 888870, 889878, 891000, 893340, 894450, 895650, 895800, 898482, 898662

There are only 168 such numbers in the range up to one million. Notice the big jumps shown in bold. The sequence jumps in a consistent manner:

• 2802 to 8010
• 8862 to 20550
• 29760 to 80472
• 88470 to 203430
• 296970 to 801000

This is best seen graphically. See Figure 1. 

Figure 1: permalink

Why such big gaps? Apart from 60, all numbers start with 2 (and end in 0 or 2) or they start with 8 (and end in 0, 2 or 8). We know that the reverse of the number must be smaller and this means that whatever digit the number starts with, the final digit must be equal to or smaller than it. This immediately rules out a number like 2093. However, why are there no numbers starting with 1, 3, 4, 5, 6, 7 or 9?

The initial composite numbers must be even as they lie between two primes so that rules out numbers whose final digits are 1, 3, 5, 7 or 9. Only composites ending in 0, 2, 4 or 8 are possible. Its reverse must also end in 0, 2, 4 or 8 if it is to lie between twin primes. If a number starts with 2 then it must end in 0 or 2. If a number starts with 4 it must end in 0, 2 or 4. If a number ends in 8 then it must end in 0, 2, 4 or 8. So why are there no numbers starting with or ending in 4? If a number starts with 4, then its reversal will end in a 4 and therefore the number above it (except for 5) cannot be prime because it ends in a 5. 

Figure 2 shows a table of the initial composites and associated primes together with the reversed composites and their associated primes.

Figure 2: permalink

Triplets of Disjoint Twin Primes

Today, having turned 26861 days, I noticed that it formed the smaller of a pair of twin primes but more than that it had a special property that qualified it for membership of OEIS A035791:

A035791

Start of a string of exactly 3 consecutive (but disjoint) pairs of twin primes.

The requirement that the three pairs be disjoint disqualifies prime pairs like (101, 103), (103, 107) and (107, 109) because of the 103 and 107 overlap. It turns out that such triplets of twin primes are relatively rare. The previous was 21587 and the next will be 49367! 

I managed to get SageMathCell to generate the OEIS sequence up to 100,000,000 as well as a table showing the record gaps between the first prime of the first prime pair and the last prime of the last prime pair. The table is shown in Figure 1 below in the range up to 100,000,000:

Figure 1: permalink

As can be seen, 26861 holds the equal record gap of 32 with the next member (49367) creating a new record of 44. The members of the sequence up to one million are shown below:

5, 179, 809, 3359, 4217, 6761, 9419, 9431, 18041, 21587, 26861, 49367, 62969, 62981, 67187, 72221, 72227, 80447, 82721, 91127, 97841, 98897, 103967, 109829, 122597, 154157, 178037, 203321, 208931, 225749, 227609, 236867, 243671, 251201, 266447, 285611, 289109, 295871, 317729, 330287, 342047, 358877, 375251, 392261, 392267, 397517, 402329, 405047, 420809, 422087, 440549, 444341, 452519, 489911, 495569, 495587, 524507, 524969, 560477, 563411, 565889, 570497, 595139, 622187, 629567, 632297, 636059, 640229, 641519, 651179, 663569, 663581, 670037, 677459, 686009, 690839, 704549, 746507, 753437, 758711, 768167, 773609, 773951, 777389, 795761, 797567, 830309, 831371, 842321, 854897, 873569, 875261, 875981, 907367, 909287, 909299, 909317, 946079, 983429, 994307, 997811

The full range of OEIS sequences are:

• A035789: Start of a string of exactly 1 consecutive (but disjoint) pairs of twin primes.
• A035790: Start of a string of exactly 2 consecutive (but disjoint) pairs of twin primes.
• A035791: Start of a string of exactly 3 consecutive (but disjoint) pairs of twin primes.
• A035792: Start of a string of exactly 4 consecutive (but disjoint) pairs of twin primes.
• A035793: Start of a string of exactly 5 consecutive (but disjoint) pairs of twin primes.
• A035794: Start of a string of exactly 6 consecutive (but disjoint) pairs of twin primes.
• A035795: Start of a string of exactly 7 consecutive (but disjoint) pairs of twin primes.

For quadruplets and beyond, the numbers are very large except for the first few members of OEIS A035792 (quadruplets) that are below a million: 9419, 62969, 72221, 392261, 495569 and 663569. 

Dream Numbers

This post is a little different from my usual content that is often prompted by an analysis of the number associated with my diurnal age. This post is prompted by a dream that I had involving a shipping container that, unlike most such containers, was gun metal in colour. It seemed totally sealed but I found a tiny opening and lit a match (or used the torch on my phone) to peer within. I saw a woman peering back at me.

She invited me in and the front face of the container disappeared so that access was now possible. She said that there were 37 people inside. Two of those were adolescent boys, one noticeably shorter than the other. It seemed that one of them was 17 but I wasn't sure which one. I pointed to the taller boy and then, quite audibly in my dream, said "seventeen?". The woman, and the mother of the two boys, smiled and clarified the situation. She said that they were twins (clearly not identical) and had been born on a Saturday. They were both 17.

After some thinking about these numbers, I realised that both 17 and 37 are 4\(k\)+1 primes and thus form the hypotenuse of right angled triangles with associated integer sides, connected by Pythagoras' Theorem:$$ \begin{align} 17^2&=8^2+15^2\\37^2&=12^2+35^2 \end{align}$$Thus we end up with a set of six numbers:$$8, 12, 15, 17, 35, 37$$Being 4\( k\)+1 primes of course, we can also write:$$ \begin{align} 17&=1^2+4^2\\37&=1^2+6^2 \end{align}$$This generates a set of five numbers:$$1, 4, 6, 17, 37$$I tend to go with the set of six numbers as they are easily associated with the Saturday night draw in the Australian lottery system. 

Being in Indonesia, I can't participate in this lottery, not even using a VPN, so I passed these numbers on to my daughter-in-law who lives in Melbourne. I suggested that she try them out. Whether she does or doesn't, I'll report back on what numbers came up in the Saturday night draw at the end of this post. I won't make the post public until after Saturday night in case anyone tries to "cash in" on my dream numbers!

LATE SATURDAY NIGHT

Well my daughter-in-law did submit the six numbers and, not surprisingly, my dream numbers did not prove precognitive. In fact I only succeeded in selecting one of the winning numbers and therefore not even a minor prize was won. See Figure 1. Nor did my other possible numbers (1, 4 and 6) make an appearance. Of course, the numbers could have been meant for Saturday April 2nd, the day before my birthday and a day that marks the 73rd solar return (when the Sun returns to the exact position that it occupied at the time of my birth).

Figure 1

Here are links to my two previous posts on the mathematics of Lotto:

• Lotto Simulations on October 20th 2018
• Lotto Loser on June 10th 2019

Of course the numbers 17 and 37 that I dreamed of may have had nothing to do with Lotto. In fact, Figure 2 gives an insight into a completely different interpretation of the numbers.

Figure 2

A shipping container is characterised by its square cross-section and so the 17 could refer to the side of this square and the 37 could refer to its length of the prism. Only two numbers are needed to define its dimensions.  When the front of the container is open and you look at it front on, the far end seems to be smaller than the open front end.  This corresponds to the twins being both 17 and yet one appearing larger than the other. 

What are the characteristics of this 17 x 17 x 37 rectangular prism? It has a volume of 10693 cubic units and a surface area of 3094 square units. If the front end is open, then the surface area is 2805 square units. How Saturday fits in with this view of things I don't know. Saturday is the sixth day of the week if we count Monday as 1, Tuesday as 2 etc. A rectangular prism does indeed have six sides.

Saturday is Saturn's day and each planet has an associated magic square. The one for Saturn is shown in Figure 3. The numbers associated with Saturn are 3, 9, 15 and 45 because the magic square is 3 x 3, there are 9 squares, the numbers in each row, column and main diagonal add to 15 and the total of all nine numbers is 45.

Figure 3

I've written about magic squares before. Here are links to these posts:

• Ultramagic Squares on November 12th 2021
• Prime Semi-Magic Squares on May 17th 2021
• Magic Squares on September 21st 2020
• Anti-Magic Squares on July 17th 2018

Other sources quote Saturn as being associated with the number 8. See Figure 4.

Figure 4: source

I seem to be going off on a tangent here so I'll stop. It has occurred to me that 17 and 37 are both reversible primes since 71 and 73 are also prime. Currently, at age 72, I'm stuck between these two primes although in less than two weeks I'll be 73. 

Let's not forget that 71 and 73 are also twin primes which fits in very nicely with the twin element in the dream. Additionally, my 73rd solar return occurs on April 2nd 2022, a Saturday. The solar return marks the return of the Sun to the same zodiacal position (12°47'07" Aries) at the time of my birth. My birthday actually occurs on April 3rd.

The shipping container may just be a symbolic representation of life on the physical plane. I am reminded of Carl Jung's description of his Near Death Experience in his autobiography "Memories, Dreams, Reflections" :

In reality, a good three weeks were still to pass before I could truly make up my mind to live again. I could not eat because all food repelled me. The view of city and mountains from my sick-bed seemed to me like a painted curtain with black holes in it, or a tattered sheet of newspaper full of photographs that meant nothing. Disappointed, I thought, "Now I must return to the 'box system' again." For it seemed to me as if behind the horizon of the cosmos a three-dimensional world had been artificially built up, in which each person sat by himself in a little box. And now I should have to convince myself all over again that this was important! Life and the whole world struck me as a prison, and it bothered me beyond measure that I should again be finding all that quite in order. I had been so glad to shed it all, and now it had come about that I along with everyone else would again be hung up in a box by a thread.

He continued:

It is impossible to convey the beauty and intensity of emotion during those visions. They were the most tremendous things I have ever experienced. And what a contrast the day was: I was tormented and on edge; everything irritated me; everything was too material, too crude and clumsy, terribly limited both spatially and spiritually. It was all an imprisonment, for reasons impossible to divine, and yet it had a kind of hypnotic power, a cogency, as if it were reality itself, for all that I had clearly perceived its emptiness. Although my belief in the world returned to returned to me, I have never since entirely freed myself of the impression that this life is a segment of existence which is enacted in a three-dimensional boxlike universe especially set up for it.

So perhaps my dream primes 17 and 37 are meant to be interpreted as the twin primes 71 and 73 with Saturday April 2nd marking my astrological coming of age 73. I realise I'm straying too far into the metaphysical here and it would be better to continue this train of thought in my blog "Mystical Meanderings".

Celebrating Gaps Between Twin Primes

Figure 1

It was back in June of 2017 that I celebrated the end of a drought of twin prime numbers (see post Gaps Between Twin Primes). I'll reproduce here, in Figure 1, the table that appears in that post. On June 22nd, I had just turned 24917 days old and 24917 is a prime that, together with 24919, forms a pair of twin primes. The previous pair is 24419 and 24421. Between the larger of the first pair (24421) and the smaller of the second pair (24917), there is a record gap of 496.

In this gap, there are 40 singleton primes or primes that are not part of a twin prime pair. This is also a record and the first of these singleton primes is 24439 that is a member of OEIS A065044, prime numbers that start a run of exactly \(n\) consecutive primes, none of which are twin primes. This remarkable gap is not surpassed until 62303, a prime that begins a run of 52 singleton primes. As can be seen in Figure 1, these 52 primes lie between the twin prime pairs of (62297, 62299) and (62927, 62929).

Today I turned 26267 days old and this number is also a member of OEIS A065044 marking a run of 36 singleton primes, the last of which is 26669. It needs to be noted that this is not a record run because the sequence registers the first prime in a run of exactly \(n\) primes. Likewise, the gap between the twin primes before and after this run of singleton primes is considerable (418) but far from the record of 496. The twin primes in question are (26681, 26683) and (26261, 26263). Figure 2 shows the situation:

Figure 2: there is a gap of 418 between the twin prime pairs

Here are the members of OEIS A065044 up to \(n\)=40:

2, 47, 113, 79, 2273, 1097, 467, 1327, 1163, 353, 5749, 3011, 5297, 10151, 1493, 9467, 887, 673, 13033, 9049, 15373, 8641, 28759, 83737, 13411, 18553, 14633, 44777, 54037, 60271, 59693, 142169, 77719, 61583, 178939, 26267, 122887, 293269, 89083, 24439

The story is not over yet however, because on the day following this post (when I turned 26268 days old) I discovered that 26268 is a member of OEIS A113274:

A113274

Record gaps between twin primes.

The members of this sequence up to 26268 are as follows:

2, 6, 12, 18, 30, 36, 72, 150, 168, 210, 282, 372, 498, 630, 924, 930, 1008, 1452, 1512, 1530, 1722, 1902, 2190, 2256, 2832, 2868, 3012, 3102, 3180, 3480, 3804, 4770, 5292, 6030, 6282, 6474, 6552, 6648, 7050, 7980, 8040, 8994, 9312, 9318, 10200, 10338, 10668, 10710, 11388, 11982, 12138, 12288, 12630, 13050, 14262, 14436, 14952, 15396, 15720, 16362, 16422, 16590, 16896, 17082, 18384, 19746, 19992, 20532, 21930, 22548, 23358, 23382, 25230, 26268, ...

However, the gap is measured from the smaller of the first twin prime pair to the larger of the second twin prime pair. For example, the gap between (17, 19) and (29, 31) is taken to be 12. Figure 1 shows the gap as being 10 because the gap is measured from the larger of first twin prime pair to the smaller of the second twin prime pair (29 - 19 = 10). Figure 1 is based on OEIS A036063 whose members differ from OEIS A113274 by 2.

A036063

Increasing gaps among twin primes: size.

All the gaps shown in Figure 1 appear in this sequence but the following shows the list of its members extended to 26266:

0, 4, 10, 16, 28, 34, 70, 148, 166, 208, 280, 370, 496, 628, 922, 928, 1006, 1450, 1510, 1528, 1720, 1900, 2188, 2254, 2830, 2866, 3010, 3100, 3178, 3478, 3802, 4768, 5290, 6028, 6280, 6472, 6550, 6646, 7048, 7978, 8038, 8992, 9310, 9316, 10198, 10336, 10666, 10708, 11386, 11980, 12136, 12286, 12628, 13048, 14260, 14434, 14950, 15394, 15718, 16360, 16420, 16588, 16894, 17080, 18382, 19744, 19990, 20530, 21928, 22546, 23356, 23380, 25228, 26266, ...

So the triplet 26266, 26267 and 26268 all make an appearance in this post. The first and last numbers are connected by the fact that the sequences to which they belong are both measuring gaps but in two different ways. The middle number is not connected with its neighbours and is measuring something quite different (the beginning of a record run of singleton primes). It's just coincidence that it falls between its two gap measuring neighbours as it does.

26262: A Special Palindrome

Indeed, "There are things that drift away, like our endless numbered days" and 26262 is one of them. Today I turned 26262 days old and this number should pass away perhaps only after it has received its due attention. 

So what makes this particular palindrome special? Well, to begin with, it's sandwiched between two primes and that doesn't happen often as OEIS A113838 reveals.

432 Hz versus 440 Hz

I've been aware for a while about the the controversy surrounding the standard A note and whether it should be set to \(440 \text{Hz} \) (as it now is) or changed to \( 432 \text{Hz}. \) I'm trying in this post to look at the mathematical properties of \( 432 \).

• \( 432^2 = 186624 \) is close to the speed of light as measured in miles per second. Wolfram Alpha gives a figure of \( 186282 \) miles per second for the speed of light in a vacuum which is \( 99.82 \text{%} \) of \( 432^2 \).
  
  
• It also turns out that the area of an equilateral triangle whose numerical area is equal to its perimeter is given by \(12 \sqrt{3} = \sqrt{432} \).
  
  
• \( 432 \) sits between the twin primes \( 431 \) and \( 433 \)
  
  
• The factors of \( 432 \) are \( 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, 216 \text{ and } 432 \). The sum of these divisors is \(1240 \).
  
  
• \(432 \) is a 3-smooth number, one that is of the form \( 2^i*3^j \text{ where }i,j>=0 \) or to put it less mathematically it is a number that can be written as a power of two times a power of three, specifically \( 2^4×3^3 \). Such numbers have been called harmonic numbers. Here are the harmonic numbers up to \( 1000 \):
  
  

• \( 432 \) is the sum of four consecutive primes: \(103+107+109+113 = 432\)
  
  
• \( 432 \) is the sum of two positive cubes: \( 6^3+6^3=432 \)
  
  
• OEIS lists \( 2944 \) entries for the number \( 432 \)

Gaps between Twin Primes

Today I turned 24917 days old. This number is prime and forms a twin prime with 24919. One claim that 24917 has to fame is that it marks the end of a gap of 496 between consecutive twin primes. After 24421 (the larger of twin pairs), there is a gap of 496 numbers until 24917 (the smaller of twin primes).

Source

At the right is a list of the record breaks. Of course, I won't live to experience the next record gap of 628 days between 62299 and 62927. However, I won't have to wait long until the next twin prime pair: 24917 and 24919. There is a gap of only 58 days between 24919 and 24977.

Of course, there's a lot of interest as to whether twin primes go on forever. It's very likely of course but it hasn't been proven. Progress has been made recently however, by Yitang Zhang, James Maynard and Terence Tao with his Polymath Project. The breakthrough relates specifically to the gaps between prime numbers rather than the gaps between successive pairs of twin prime numbers. There's a good account of these developments in this article from quantamagazine.

In any case, I felt it was important to celebrate the end of this latest record gap as my endless, numbered days unravel.

Gaps between Twin Primes

Today is a prime day for me, the prime being the larger half of the twin prime pair 24419 and 24421. Just as the gaps between primes are variable, so too are the gaps between successive twin primes. However, at certain points records are set regarding how big these gaps are. It just so happens that 24421 marks one of those points. The next prime pair is 24917 and 24919, and the gap of 496 between 24421 and 24917 sets a record. A list of the initial record intervals is attached.

24421 is a member of OEIS A036061: increasing gaps among twin primes, the largest prime of the starting twin pair. The record gaps between primes was treated in this earlier post.

Double and Reverse Digits

After twelve days, I encountered today the first member of the twin prime pair: 24371 and 24373. It's been a while: the last pair was 24179 and 24181 as far as I can tell. The number is a member of the interesting OEIS A036447 formed using 1 as its starting point and then doubling and reversing the digits:

1, 2, 4, 8, 61, 221, 244, 884, 8671, 24371, ...

The number is also a member of OEIS A243408: primes p such that 10p-1, 10p-3, 10p-7 and 10p-9 are all prime. This means that 243709, 243707, 243703 and 243701 are all prime.

Additionally, the number is a member of OEIS A158641: strong primes p: adding 2 to any one digit of p produces a prime number (no digits 8 & 9 in p). This means that 44371, 26371, 24571, 24391 and 24373 are prime.

There's still more. The number is a member of OEIS A104846: primes from merging of 5 successive digits in decimal expansion of e. Here is part of the sequence (up to 24371):

74713, 62497, 24977, 24709, 47093, 95957, 49669, 27427, 46639, 32003, 59921, 21817, 35729, 63073, 28627, 27943, 94349, 33829, 98807, 57383, 41879, 18793, 91499, 68477, 47741, 37423, 42437, 24371

Lastly, 24371 is also a member of OEIS A054564 as describe below: