Palindions
Math TriviaA palindrome is a number that reads the same way forward and backward. Examples are 1234321, 484, and 6300036. A number’s divisors are all the numbers that divide evenly into it. The divisors of 10, for example, are 1, 2, 5, and 10 because 10/1 = 10, 10/2 = 5, 10/5 = 2, and 10/10 = 1. Let’s combine the concepts of palindromes and divisors: What would the sequence of numbers that possess an abundance of palindromic divisors look like?
1, 2, 4, 6, 12, 24, 66, 132, 264, 792, 1848, 2772, 5544, 13332, 14652, 24024, 26664, 72072, 79992, 186648, 205128, 264264, 559944, 792792, 1333332, 2666664, 7279272, 7999992, 13333320, 14666652, 26690664, 29333304, 80071992, 134666532
A formal definition of this sequence is: Numbers n such that the amount of palindromic divisors of n sets a new record. In Clifford A. Pickover’s book, A Passion For Mathematics, he dubbed these numbers “palindions” and wrote: “Palindions are natural numbers that have more palindromic divisors than any smaller number” (p. 107). For example, 2666664 is a palindion because it has exactly 50 palindromic divisors, and no smaller number has that many. (The sequence above is listed as A093036 in Neil Sloane’s Online Encyclopedia of Integer Sequences.)
If we look at the actual divisors of the palindions, usually we will see the palindromic divisors scattered “randomly” among the others, but with many making up the larger divisors. Here are all divisors of 24024, for example: [1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 21, 22, 24, 26, 28, 33, 39, 42, 44, 52, 56, 66, 77, 78, 84, 88, 91, 104, 132, 143, 154, 156, 168, 182, 231, 264, 273, 286, 308, 312, 364, 429, 462, 546, 572, 616, 728, 858, 924, 1001, 1092, 1144, 1716, 1848, 2002, 2184, 3003, 3432, 4004, 6006, 8008, 12012, 24024].
Notice 8008, 6006, 4004, 3003, 2002, and 1001 among the larger divisors.
The palindion 250128, however, is somewhat of an anomaly in this regard. Here are its divisors: [1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 18, 21, 22, 24, 28, 33, 36, 37, 42, 44, 56, 63, 66, 72, 74, 77, 84, 88, 99, 111, 126, 132, 148, 154, 168, 198, 222, 231, 252, 259, 264, 296, 308, 333, 396, 407, 444, 462, 504, 518, 616, 666, 693, 777, 792, 814, 888, 924, 1036, 1221, 1332, 1386, 1554, 1628, 1848, 2072, 2331, 2442, 2664, 2772, 2849, 3108, 3256, 3663, 4662, 4884, 5544, 5698, 6216, 7326, 8547, 9324, 9768, 11396, 14652, 17094, 18648, 22792, 25641, 29304, 34188, 51282, 68376, 102564, 205128]
Notice it has fewer palindromic divisors in the higher register. Is there a reason the palindromic divisors of 250128 are distributed differently than the other palindions?
Robert G. Wilson made the observation that all palindions greater than 132 are divisible by 11*12 = 132, with the factor 11 serving to generate palindromic divisors, and 12 producing a larger amount of them. Does this mean we can take a palindion and continue multiplying it by 132 to get numbers with increasingly larger amounts of palindromic divisors? Not necessarily. Let x = 14666652 and npd(x) the number of palindromic divisors of x; npd(x) = 59, npd(x * 132) = 82, npd(x * 1322) = 101, npd(x * 1323) = 110, but npd(x * 1324) = 110 once again.
However, there is a way to find numbers with larger amounts of palindromic divisors; and 132, 11, and 12 are each important to the process. Notice in the palindion sequence the terms 132, 13332, and 1333332 occur. If we let f(n) = 4/3 * (10n -1) it produces this class of numbers, and we can check to see how many palindromic divisors each term has.
npd(f(n)) for n = 1 to 25:
5, 10, 11, 20, 10, 43, 11, 42, 23, 34, 10, 133, 13, 37, 36, 84, 10, 153, 10, 107, 40, 43, 10, 426, 20.
While not steadily increasing, notice the larger amounts of palindromic divisors occur when n = 12 and 24. Will multiples of 12 continue to produce more and more palindromic divisors?
npd(f(12)) = 133
npd(f(24)) = 426
npd(f(36)) = 674
npd(f(48)) = 1297
1297 is a lot of palindromic divisors! I attempted to compute npd(f(60)) but my computer did not have enough memory.
Open questions: Why does f(n) = 4/3 * (10n -1) continue to produce larger amounts of palindromic divisors whenever n is a multiple of 12? Will those multiples continue to increase?
Also notice in the original palindion sequence the terms 264, 26664, and 2666664 occur. The formula for these numbers is of course simply f(n)*2. When we let g(n) = f(n)*2 and compute npd(g(n * 12)) it also yields numbers with a high amount of palindromic divisors. It seems that f(n)*3, f(n)*4, in general f(n)*k continues to produce numbers with higher amounts of palindromic divisors when n is a multiple of 12; and further computation reveals that when k is a multiple of 11 it seems to yield even large amounts! – so we are back to the original 11*12 observation.
Challenge: What is the smallest number you can find that has the largest amount of palindromic divisors? The smallest I have found is
(f(48)*11*80)
= 1173333333333333333333333333333333333333333333332160
which has 258048 total divisors, with 2448 of them palindromic.
Now I will close this article by listing the divisors of a number that has (in my opinion) many aesthetically pleasing digital patterns. (The number could possibly be a term of the palindion sequence, but I have not computed the full sequence to its level yet.) Here are the divisors of 13333333333333332:
[1, 2, 3, 4, 6, 11, 12, 17, 22, 33, 34, 44, 51, 66, 68, 73, 101, 102, 132, 137, 146, 187, 202, 204, 219, 274, 292, 303, 374, 404, 411, 438, 548, 561, 606, 748, 803, 822, 876, 1111, 1122, 1212, 1241, 1507, 1606, 1644, 1717, 2222, 2244, 2329, 2409, 2482, 3014, 3212, 3333, 3434, 3723, 4444, 4521, 4658, 4818, 4964, 5151, 6028, 6666, 6868, 6987, 7373, 7446, 9042, 9316, 9636, 10001, 10302, 13332, 13651, 13837, 13974, 14746, 14892, 18084, 18887, 20002, 20604, 22119, 25619, 27302, 27674, 27948, 29492, 30003, 37774, 40004, 40953, 41511, 44238, 51238, 54604, 55348, 56661, 60006, 75548, 76857, 81103, 81906, 83022, 88476, 102476, 110011, 113322, 120012, 125341, 152207, 153714, 162206, 163812, 166044, 170017, 220022, 226644, 235229, 243309, 250682, 304414, 307428, 324412, 330033, 340034, 376023, 440044, 456621, 470458, 486618, 501364, 510051, 608828, 660066, 680068, 705687, 752046, 913242, 940916, 973236, 1010101, 1020102, 1320132, 1378751, 1411374, 1504092, 1826484, 1870187, 2020202, 2040204, 2587519, 2757502, 2822748, 3030303, 3740374, 4040404, 4136253, 5175038, 5515004, 5610561, 5882353, 6060606, 7480748, 7762557, 8272506, 10350076, 11111111, 11221122, 11764706, 12121212, 15525114, 16545012, 17171717, 17647059, 22222222, 22442244, 23529412, 31050228, 33333333, 34343434, 35294118, 44444444, 51515151, 64705883, 66666666, 68686868, 70588236, 100000001, 103030302, 129411766, 133333332, 188888887, 194117649, 200000002, 206060604, 258823532, 300000003, 377777774, 388235298, 400000004, 429411769, 566666661, 594117653, 600000006, 755555548, 776470596, 805882361, 858823538, 1100000011, 1133333322, 1188235306, 1200000012, 1288235307, 1611764722, 1717647076, 1782352959, 2200000022, 2266666644, 2376470612, 2417647083, 2576470614, 3223529444, 3300000033, 3564705918, 4400000044, 4723529459, 4835294166, 5152941228, 6535294183, 6600000066, 7129411836, 7300000073, 8864705971, 9447058918, 9670588332, 10100000101, 13070588366, 13200000132, 13700000137, 14170588377, 14600000146, 17729411942, 18894117836, 19605882549, 20200000202, 21900000219, 26141176732, 26594117913, 27400000274, 28341176754, 29200000292, 30300000303, 35458823884, 39211765098, 40400000404, 41100000411, 43370588669, 43800000438, 53188235826, 54800000548, 56682353508, 58829412353, 60600000606, 78423530196, 80300000803, 81394118461, 82200000822, 86741177338, 87600000876, 106376471652, 111100001111, 117658824706, 121200001212, 130111766007, 150700001507, 160600001606, 162788236922, 164400001644, 173482354676, 176488237059, 222200002222, 235317649412, 240900002409, 244182355383, 260223532014, 301400003014, 321200003212, 325576473844, 333300003333, 352976474118, 444400004444, 452100004521, 477076475359, 481800004818, 488364710766, 520447064028, 602800006028, 647123535883, 666600006666, 705952948236, 737300007373, 895335303071, 904200009042, 954152950718, 963600009636, 976729421532, 1000100010001, 1294247071766, 1333200013332, 1383700013837, 1431229426077, 1474600014746, 1790670606142, 1808400018084, 1908305901436, 1941370607649, 2000200020002, 2211900022119, 2588494143532, 2686005909213, 2767400027674, 2862458852154, 2949200029492, 3000300030003, 3581341212284, 3882741215298, 4000400040004, 4151100041511, 4423800044238, 5372011818426, 5534800055348, 5724917704308, 5941770647653, 6000600060006, 7765482430596, 8110300081103, 8302200083022, 8847600088476, 10744023636852, 11001100110011, 11883541295306, 12001200120012, 15220700152207, 16220600162206, 16604400166044, 17825311942959, 22002200220022, 23767082590612, 24330900243309, 30441400304414, 32441200324412, 33003300330033, 35650623885918, 44004400440044, 45662100456621, 48661800486618, 60882800608828, 65359477124183, 66006600660066, 71301247771836, 91324200913242, 97323600973236, 101010101010101, 130718954248366, 132013201320132, 182648401826484, 196078431372549, 202020202020202, 261437908496732, 303030303030303, 392156862745098, 404040404040404, 606060606060606, 784313725490196, 1111111111111111, 1212121212121212, 2222222222222222, 3333333333333333, 4444444444444444, 6666666666666666, 13333333333333332]
