Palindromes In Two Or More Consecutive Number Bases

Having turned a venerable 26457 days old, I searched through the OEIS to find an interesting sequence in which this number appeared. Nothing caught my fancy but I did stumble upon an interesting post to the Mathematics section of StackExchange. See Figure 1.

Figure 1

An extended search within the OEIS, looking for so-called b-files (see my December 2019 article titled In the Groove), had revealed nothing of interest but Google had returned a result because the number appeared in the StackExchange post and that post contained links to the OEIS. The post began as follows:

Lets look at sequences of numbers that are a palindromic number in two consecutive number bases $b$ and $𝑏+1$, where $𝑏 \geq 2$ of course. (And also ignoring trivial one digit palindromes.)

I would conjecture that there are infinitely many numbers palindromic in two consecutive number bases for any two number bases $(𝑏,𝑏+1)$ where $b \geq 2$. But I do not know how to show that this statement is true.

Turns out, it is in fact not known if the case $(2,3)$ for example, has infinitely many terms, since the OEIS entry for it is written as "...if it exists". There are no clear patterns in this particular sequence, as it seems.

This is the OEIS entry referred to:

A060792

Numbers that are palindromic in bases 2 and 3.

The first few entries are: 0, 1, 6643, 1422773, 5415589, 90396755477, ... so such numbers are relatively sparse. Looking at the first non-trivial number in this sequence (6643), we see that in base 2, 6643 --> 1100111110011 and in base 3, 6643 --> 100010001.

The StackExchange goes on to list numbers that are palindromic is two consecutive number bases (with $b$ ranging from 2 to 36):

Numbers up to 107 in number bases up to 32 (where the ∗ indicates that the number is also palindromic in a third consecutive base) :

(2, 3): 6643, 1422773, 5415589, ... OEIS A060792

(3, 4): 10, 130, 11950, 175850, 749470, 1181729, ... OEIS A097928

(4, 5): 46, 9222, 76449, 193662, 2347506, 2593206, ... OEIS A097929

(5, 6): 67, 98, 104, 651, 2293, 3074, 26691, 27741, 704396, 723296, 755846, 883407, ... OEIS A097930

(6, 7): 92, 135, *178, 185, 5854, 6148, 7703, 186621, 204856, 206620, 213970, 269957, 271721, 279071, ... OEIS A097931

(7, 8): 121, 178, 235, 292, *300, 2997, 6953, 7801, 10658, 13459, 16708, 428585, 431721, 444713, 447849, ... OEIS A099145

(8, 9): 154, 227, 300, *373, 446, 455, 11314, 12547, 17876, 27310, 889435, 894619, 899803, 926371, ... OEIS A099146

(9, 10): 191, 282, 373, 464, 555, 646, 656, 6886, 25752, 27472, 42324, 50605, 626626, 1540451, 1713171, 1721271, 1828281, 1877781, 1885881, 2401042, 2434342, 2442442, 2450542, 3106013, 3114113, 3122213, 3163613, 3171713, 3303033, *3360633, ... OEIS A029965

(10, 11): 232, 343, 454, 565, *676, 787, 898, 909, 26962, 38183, 40504, 49294, 52825, 63936, 75157, 2956592, 2968692, 3262623, 3274723, 3286823, 3298923, 3360633, 3372733, 4348434, 4410144, 4422244, 4581854, 4593954, 5643465, 5655565, 5667665, 5741475, 7280827, 7292927, 8710178, 8722278, 8734378, 8746478, 8758578, 8820288, 8832388, ... OEIS A029966

(11, 12): 277, 410, 543, 676, 809, 942, 1075, 1208, 1220, 38425, 54662, 72351, 75399, 93088, 125430, 1798303, 1817179, 5058385, 5075809, 5093233, 5199361, 5216785, 5550889, 5568313, 5585737, 5603161, 5620585, 7569434, 7727702, 7833830, 7851254, 7868678, 7886102, 9711399, 9728823, 9746247, 

(12, 13): 326, 483, 640, 797, 954, *1111, 1268, 1425, 1582, 1595, 53210, 100636, 104549, 123257, 129198, 151819, 174596, 227806, 8281118, 8305454, 8329790, 8354126, 8502170, 8526506, 9041475, 9065811, 9090147, 9114483, 

(13, 14): 379, 562, 745, 928, 1111, 1294, 1477, 1660, 1843, 2026, 2040, 71905, 105394, 136517, 167458, 170006, 174934, 205875, 208423, 239364, 270487, 342392, 344954, 

(14, 15): 436, 647, 858, 1069, 1280, 1491, *1702, 1913, 2124, 2335, 2546, 2561, 95146, 139667, 181248, 225769, 231874, 267140, 276395, 317766, 454493, 499014, 502179, 

(15, 16): 497, 738, 979, 1220, 1461, 1702, 1943, 2184, 2425, 2666, 2907, 3148, 3164, 123617, 181698, 294260, 348501, 359797, 414038, 472119, 526600, 650217, 708298, 712154, 

(16, 17): 562, 835, 1108, 1381, 1654, 1927, 2200, *2473, 2746, 3019, 3292, 3565, 3838, 3855, 158050, 232579, 307108, 377285, 447190, 451814, 460807, 521719, 530712, 535336, 605241, 679770, 833468, 907997, 977902, 982526, 987167, 

(17, 18): 631, 938, 1245, 1552, 1859, 2166, 2473, 2780, 3087, 3394, 3701, 4008, 4315, 4622, 4640, 199225, 293474, 387723, 476770, 571019, 659760, 675996, 764737, 858986, 1147258, 1241507, 1330248, 1341282, 

(18, 19): 704, 1047, 1390, 1733, 2076, 2419, 2762, 3105, *3448, 3791, 4134, 4477, 4820, 5163, 5506, 5525, 247970, 365619, 483268, 712410, 823561, 842732, 941210, 953883, 1071532, 1189181, 1430995, 1548644, 1666293, 1777444, 

(19, 20): 781, 1162, 1543, 1924, 2305, 2686, 3067, 3448, 3829, 4210, 4591, 4972, 5353, 5734, 6115, 6496, 6516, 305161, 450322, 595483, 740644, 878585, 1016146, 1023746, 1161307, 1176147, 1183747, 1321308, 1466469, 1611630, 1909571, 2054732, 2192293, 2199893, 2337454, 2352294, 

(20, 21): 862, 1283, 1704, 2125, 2546, 2967, 3388, 3809, 4230, *4651, 5072, 5493, 5914, 6335, 6756, 7177, 7598, 7619, 371722, 548963, 726204, 903445, 1072286, 1249527, 1417948, 1444009, 1595189, 1612430, 1789671, 1966912, 2330234, 2507475, 2684716, 2853137, 3030378, 3047619, 

(21, 22): 947, 1410, 1873, 2336, 2799, 3262, 3725, 4188, 4651, 5114, 5577, 6040, 6503, 6966, 7429, 7892, 8355, 8818, 8840, 448625, 662994, 877363, 1091732, 1510768, 1714973, 1929342, 1949230, 2163599, 2377968, 2592337, 3031260, 3245629, 3459998, 3664203, 3878572, 3898460, 

(22, 23): 1036, 1543, 2050, 2557, 3064, 3571, 4078, 4585, 5092, 5599, *6106, 6613, 7120, 7627, 8134, 8641, 9148, 9655, 10162, 10185, 536890, 793939, 1050988, 1308037, 1565086, 1811003, 2056414, 2068052, 2313463, 2347894, 2570512, 2593305, 2850354, 3107403, 3364452, 3633161, 3890210, 4147259, 4392670, 4404308, 4649719, 4906768, 4929561, 

(23, 24): 1129, 1682, 2235, 2788, 3341, 3894, 4447, 5000, 5553, 6106, 6659, 7212, 7765, 8318, 8871, 9424, 9977, 10530, 11083, 11636, 11660, 637585, 943394, 1249203, 1555012, 1860821, 2153934, 2459743, 2752304, 3058113, 3084081, 3389890, 3695699, 4001508, 4626397, 4932206, 5238015, 5530576, 5836385, 6142194, 6168162, 

(24, 25): 1226, 1827, 2428, 3029, 3630, 4231, 4832, 5433, 6034, 6635, 7236, *7837, 8438, 9039, 9640, 10241, 10842, 11443, 12044, 12645, 13246, 13271, 751826, 1113027, 1474228, 1835429, 2196630, 2904632, 3250833, 3612034, 3973235, 4002660, 4363861, 4725062, 5086263, 5462488, 5823689, 6184890, 6546091, 6892292, 7253493, 7614694, 7644119, 

(25, 26): 1327, 1978, 2629, 3280, 3931, 4582, 5233, 5884, 6535, 7186, 7837, 8488, 9139, 9790, 10441, 11092, 11743, 12394, 13045, 13696, 14347, 14998, 15024, 880777, 1304578, 1728379, 2152180, 2575981, 2999782, 3407333, 3814234, 3831134, 4238035, 4661836, 4695012, 5118813, 5542614, 5966415, 6830942, 7254743, 7678544, 8085445, 8102345, 8509246, 8933047, 9356848, 9390024, 

(26, 27): 1432, 2135, 2838, 3541, 4244, 4947, 5650, 6353, 7056, 7759, 8462, 9165, *9868, 10571, 11274, 11977, 12680, 13383, 14086, 14789, 15492, 16195, 16898, 16925, 1025650, 1519859, 2014068, 2508277, 3002486, 3496695, 3972652, 4466861, 4942116, 5436325, 5930534, 5967767, 6461976, 6956185, 7450394, 7963583, 8457792, 8952001, 9446210, 9921465, 

(27, 28): 1541, 2298, 3055, 3812, 4569, 5326, 6083, 6840, 7597, 8354, 9111, 9868, 10625, 11382, 12139, 12896, 13653, 14410, 15167, 15924, 16681, 17438, 18195, 18952, 18980, 1187705, 1760754, 2333803, 2906852, 3479901, 4052950, 5178636, 5730517, 6303566, 6876615, 6918223, 7491272, 8064321, 8637370, 9804663, 

(28, 29): 1654, 2467, 3280, 4093, 4906, 5719, 6532, 7345, 8158, 8971, 9784, 10597, 11410, *12223, 13036, 13849, 14662, 15475, 16288, 17101, 17914, 18727, 19540, 20353, 21166, 21195, 1368250, 2029219, 2690188, 3351157, 4012126, 4673095, 5334064, 5972297, 6609718, 6633266, 7270687, 7931656, 8592625, 8638938, 9299907, 9960876, 

(29, 30): 1771, 2642, 3513, 4384, 5255, 6126, 6997, 7868, 8739, 9610, 10481, 11352, 12223, 13094, 13965, 14836, 15707, 16578, 17449, 18320, 19191, 20062, 20933, 21804, 22675, 23546, 23576, 1568641, 2327282, 3085923, 3844564, 4603205, 5361846, 6120487, 6853898, 7612539, 8345080, 9103721, 9862362, 9913722, 

(30, 31): 1892, 2823, 3754, 4685, 5616, 6547, 7478, 8409, 9340, 10271, 11202, 12133, 13064, 13995, *14926, 15857, 16788, 17719, 18650, 19581, 20512, 21443, 22374, 23305, 24236, 25167, 26098, 26129, 1790282, 2657043, 3523804, 4390565, 5257326, 6124087, 6990848, 8696470, 9534401, 

(31, 32): 2017, 3010, 4003, 4996, 5989, 6982, 7975, 8968, 9961, 10954, 11947, 12940, 13933, 14926, 15919, 16912, 17905, 18898, 19891, 20884, 21877, 22870, 23863, 24856, 25849, 26842, 27835, 28828, 28860, 2034625, 3020674, 4006723, 4992772, 5978821, 6964870, 7950919, 8936968, 9892265, 

(32, 33): 2146, 3203, 4260, 5317, 6374, 7431, 8488, 9545, 10602, 11659, 12716, 13773, 14830, 15887, 16944, *18001, 19058, 20115, 21172, 22229, 23286, 24343, 25400, 26457, 27514, 28571, 29628, 30685, 31742, 31775, 2303170, 3420419, 4537668, 5654917, 6772166, 7889415, 9006664, 

Right at the very end of the above list, under number bases $b=32$ and $b=33$, we find $26457$.

$$\begin{align} 26457_{_{10}}&=\text{ pqp }_{_{32}}\\&=\text{ o9o }_{_{33}} \end{align}$$ What this means is that with $o=24, p=25, q=26$ we have: $$\begin{align} 26457 &=25 \times 32^2+26 \times 32+25\\ &=24 \times 33^2+9 \times 33+24 \end{align}$$ These numbers in fact form OEIS A279092:

A279092

Numbers that are nontrivially palindromic in two or more consecutive integer bases.

The initial members are:

10, 46, 67, 92, 98, 104, 121, 130, 135, 154, 178, 185, 191, 227, 232, 235, 277, 282, 292, 300, 326, 343, 373, 379, 410, 436, 446, 454, 455, 464, 483, 497, 543, 555, 562, 565, 631, 640, 646, 647, 651, 656, 676, 704, 738, 745, 781, 787, 797, 809, 835, 858, 862

Notice that in the StackExchange list for $b=32$ and $b=33$, $18001$ is marked with an asterisk and so $b=34$ is included as well:

$$\begin{align} 18001_{_{10}}&=\text{ hih }_{_{32}}\\&=\text{ ghg }_{_{33}} \\&=\text{ fjf }_{_{34}} \end{align}$$ Extracting the numbers with asterisks in the above list (remember the ∗ indicates that the number is also palindromic in a third consecutive base), we get:

(06, 07): 178 --> 454 b=6, 343 b=7, 262 b=8

(07, 08): 300 --> 606 b=7, 454 b=8, 363 b=9

(08, 09): 373 --> 565 b=8, 454 b=9, 373 b=10

(09, 10): 3360633 --> 6281826 b=9, 3360633 b=10 1995991 b=11

(10, 11): 676 --> 676 b=10, 565 b=11, 484 b=12

(12, 13): 1111 --> 787 b=12, 676 b=13, 595 b=14

(14, 15): 1702 --> 898 b=14, 787 b=15, 6a6 b=16

(16, 17): 2473 --> 9a9 b=16, 898 b=17, 7b7 b=18

(18, 19): 3448 --> aba b=18, 9a9 b=19, 8c8 b=20

(20, 21): 4651 --> bcb b=20, aba b=21, 9d9 b=22

(22, 23): 6106 --> cdc b=22, bcb b=23, aea b=24

(24, 25): 7837 --> ded base=24, cdc b=25, bfb b=26

(26, 27): 9868 --> efe b=26, ded b=27, cgc b=28

(28, 29): 12223 --> fgf b=28, efe b=29, dhd b=30

(30, 31): 14926 --> ghg b=30, fgf b=31, eie b=32

(32, 33): 18001 --> hih b=32, ghg b=33, fjf b=34

These numbers make up OEIS A279093:

A279093

Numbers that are nontrivially palindromic in three or more consecutive integer bases.

The initial members are:

178, 300, 373, 676, 1111, 1702, 2473, 3448, 4651, 6106, 7837, 9868, 12223, 14926, 18001, 21472, 25363, 29698, 34501, 39796, 45607, 51958, 58873, 66376, 74491, 83242, 92653, 102748, 113551, 125086, 137377, 150448, 164323, 179026, 194581, 211012, 228343, 246598

Here is a list of the first thousand such numbers. 

The comments to the OEIS entry include the statement that no numbers have been found that are palindromes in four successive bases. 130 is given as an example of a number that is palindromic in seven integer bases:

$$11211_3 = 2002_4 = 202_8 = {\large aa}_{12} = 55_{25} = 22_{64} = 11_{129}$$ but these bases do not include three consecutive integers, so 130 is not in the sequence. Incidentally, I'm creating this palindromic post on the 9th September 2021 and on the 12th September, the date can be written palindromically as 12/9/21.