Monday, 9 June 2025

Nothing New Under The Sun

Every day I try to form a digit equation from the digits of the number associated with my diurnal age. Today's number was 27826. Usually I succeed but sometimes I don't. I've described the rules in my blog post titled Forming Digit Equations: A Game. Applying these rules to 27826 we get (with | representing divided into):2+7=82|65=83=5Looking at this number however, something else caught my eye and it involved powers of 2. I thought: suppose that instead of 27 there was 27=128 and instead of 26 there was 26=64. In that case the number would be 128864. Now we have a number that is a concatenation of powers of 2 because (with | this time representing concatenation) we have:128864=27|23|26I wondered how many numbers in the range up to 40000 have the property that they are a concatenation of powers of 2, which includes 20=1. As is my lazy way nowadays I put Gemini to work writing the Python code (permalink) to find all the numbers in this range. It turns out that there are 1563 such numbers with the largest being 32888 where:32888=25|23|23|23The initial numbers are 1, 2, 4, 8, 11, 12, 14, 16, 18, 21 and so I thought I'd put these numbers into the OEIS and see if anything turned up. To my surprise, the sequence is listed (A381259). See Figure 1.

Figure 1: link

Hence the title of this post "Nothing New Under the Sun" because clearly someone had thought of such a sequence before I had. I won't list all the numbers in the range up to 40000 but here are the ones that are coming up for me:

28111, 28112, 28114, 28116, 28118, 28121, 28122, 28124, 28128, 28132, 28141, 28142, 28144, 28148, 28161, 28162, 28164, 28168, 28181, 28182, 28184, 28188, 28192, 28211, 28212, 28214, 28216, 28218, 28221, 28222, 28224, 28228, 28232, 28241, 28242, 28244, 28248, 28256, 28264, 28281, 28282, 28284, 28288, 28321, 28322, 28324, 28328, 28411, 28412, 28414, 28416, 28418, 28421, 28422, 28424, 28428, 28432, 28441, 28442, 28444, 28448, 28464, 28481, 28482, 28484, 28488, 28512, 28641, 28642, 28644, 28648, 28811, 28812, 28814, 28816, 28818, 28821, 28822, 28824, 28828, 28832, 28841, 28842, 28844, 28848, 28864, 28881, 28882, 28884, 28888, 32111, 32112, 32114, 32116, 32118, 32121, 32122, 32124, 32128, 32132, 32141, 32142, 32144, 32148, 32161, 32162, 32164, 32168, 32181, 32182, 32184, 32188, 32211, 32212, 32214, 32216, 32218, 32221, 32222, 32224, 32228, 32232, 32241, 32242, 32244, 32248, 32256, 32264, 32281, 32282, 32284, 32288, 32321, 32322, 32324, 32328, 32411, 32412, 32414, 32416, 32418, 32421, 32422, 32424, 32428, 32432, 32441, 32442, 32444, 32448, 32464, 32481, 32482, 32484, 32488, 32512, 32641, 32642, 32644, 32648, 32768, 32811, 32812, 32814, 32816, 32818, 32821, 32822, 32824, 32828, 32832, 32841, 32842, 32844, 32848, 32864, 32881, 32882, 32884, 32888

I looked long and hard at 32768 in the list above because I couldn't see how it could be a concatenation of powers of 2. I had to ask Gemini to clarify. It did:32768=215So there you have it. I thought I'd discovered a brand new sequence but it was already in the OEIS database but it hadn't been there for long as the date of inclusion is February 18th 2025.

Now the Python code is easily modified to deal with the powers of other numbers. Let's consider powers of 3. There are 570 numbers satisfying this criterion in the range up to 40000 (permalink). Here are the ones that are coming up for me (some of which I'll see hopefully):

27911, 27913, 27919, 27927, 27931, 27933, 27939, 27981, 27991, 27993, 27999, 31111, 31113, 31119, 31127, 31131, 31133, 31139, 31181, 31191, 31193, 31199, 31243, 31271, 31273, 31279, 31311, 31313, 31319, 31327, 31331, 31333, 31339, 31381, 31391, 31393, 31399, 31729, 31811, 31813, 31819, 31911, 31913, 31919, 31927, 31931, 31933, 31939, 31981, 31991, 31993, 31999, 32187, 32431, 32433, 32439, 32711, 32713, 32719, 32727, 32731, 32733, 32739, 32781, 32791, 32793, 32799, 33111, 33113, 33119, 33127, 33131, 33133, 33139, 33181, 33191, 33193, 33199, 33243, 33271, 33273, 33279, 33311, 33313, 33319, 33327, 33331, 33333, 33339, 33381, 33391, 33393, 33399, 33729, 33811, 33813, 33819, 33911, 33913, 33919, 33927, 33931, 33933, 33939, 33981, 33991, 33993, 33999, 36561, 37291, 37293, 37299, 38111, 38113, 38119, 38127, 38131, 38133, 38139, 38181, 38191, 38193, 38199, 39111, 39113, 39119, 39127, 39131, 39133, 39139, 39181, 39191, 39193, 39199, 39243, 39271, 39273, 39279, 39311, 39313, 39319, 39327, 39331, 39333, 39339, 39381, 39391, 39393, 39399, 39729, 39811, 39813, 39819, 39911, 39913, 39919, 39927, 39931, 39933, 39939, 39981, 39991, 39993, 39999

Let's take the first number in the list above:27911=33|32|30|30Similarly with powers of 5 where we have 103 suitable numbers in the range up to 40000 (permalink). We can list them here as there aren't that many.

1, 5, 11, 15, 25, 51, 55, 111, 115, 125, 151, 155, 251, 255, 511, 515, 525, 551, 555, 625, 1111, 1115, 1125, 1151, 1155, 1251, 1255, 1511, 1515, 1525, 1551, 1555, 1625, 2511, 2515, 2525, 2551, 2555, 3125, 5111, 5115, 5125, 5151, 5155, 5251, 5255, 5511, 5515, 5525, 5551, 5555, 5625, 6251, 6255, 11111, 11115, 11125, 11151, 11155, 11251, 11255, 11511, 11515, 11525, 11551, 11555, 11625, 12511, 12515, 12525, 12551, 12555, 13125, 15111, 15115, 15125, 15151, 15155, 15251, 15255, 15511, 15515, 15525, 15551, 15555, 15625, 16251, 16255, 25111, 25115, 25125, 25151, 25155, 25251, 25255, 25511, 25515, 25525, 25551, 25555, 25625, 31251, 31255

Let's take the last number in the list:31255=55|51Representations are not necessarily unique. For example:15125=50|51|53=50|51|50|52Finally let's consider powers of 7 where there are 96 in the range up to 40000 (permalink). They are:

1, 7, 11, 17, 49, 71, 77, 111, 117, 149, 171, 177, 343, 491, 497, 711, 717, 749, 771, 777, 1111, 1117, 1149, 1171, 1177, 1343, 1491, 1497, 1711, 1717, 1749, 1771, 1777, 2401, 3431, 3437, 4911, 4917, 4949, 4971, 4977, 7111, 7117, 7149, 7171, 7177, 7343, 7491, 7497, 7711, 7717, 7749, 7771, 7777, 11111, 11117, 11149, 11171, 11177, 11343, 11491, 11497, 11711, 11717, 11749, 11771, 11777, 12401, 13431, 13437, 14911, 14917, 14949, 14971, 14977, 16807, 17111, 17117, 17149, 17171, 17177, 17343, 17491, 17497, 17711, 17717, 17749, 17771, 17777, 24011, 24017, 34311, 34317, 34349, 34371, 34377

Let's take the last number in the list again:34377=73|71|71Apart from the powers of 2, none of the sequences above are listed in the OEIS. Of course, those numbers containing only the digit 1 are common to all the sequences.

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