Every day I try to form a digit equation from the digits of the number associated with my diurnal age. Today's number was
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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:
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:
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:
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:
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