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%I #54 Feb 08 2019 06:27:09
%S 178,300,373,676,1111,1702,2473,3448,4651,6106,7837,9868,12223,14926,
%T 18001,21472,25363,29698,34501,39796,45607,51958,58873,66376,74491,
%U 83242,92653,102748,113551,125086,137377,150448,164323,179026,194581,211012,228343,246598
%N Numbers that are nontrivially palindromic in three or more consecutive integer bases.
%C For any integer b > 1, the base-b expansion of any number k < b will be a one-digit number, and will thus be trivially palindromic.
%C For each j >= 5 and odd, k = (j^3 + 6*j^2 + 14*j + 11)/2 is a term in the sequence, and represents a 3-digit palindrome in each of three consecutive integer bases:
%C .
%C base 1st digit 2nd digit 3rd digit
%C ---- --------- --------- ---------
%C j+1 (j+3)/2 (j+5)/2 (j+3)/2
%C j+2 (j+1)/2 (j+3)/2 (j+1)/2
%C j+3 (j-1)/2 (j+7)/2 (j-1)/2
%C .
%C (see 178 and 373 in the Example section). Nearly all of the first 95 terms of this sequence are terms of this form.
%C For each j >= 44 and divisible by 4, k = (3*j^5 + 30*j^4 + 125*j^3 + 270*j^2 + 307*j + 148)/4 is a term in the sequence, and represents a 5-digit palindrome in each of three consecutive integer bases:
%C .
%C base 1st digit 2nd digit 3rd digit 4th digit 5th digit
%C ---- --------- --------- --------- --------- ---------
%C j+1 3*j/4 + 4 j/2 + 9 j/4 + 11 j/2 + 9 3*j/4 + 4
%C j+2 3*j/4 + 1 j/2 + 2 j/4 + 0 j/2 + 2 3*j/4 + 1
%C j+3 3*j/4 - 2 j/2 + 10 j/4 - 11 j/2 + 10 3*j/4 - 2
%C .
%C [Reformatted by _Jon E. Schoenfield_, Apr 01 2018]
%C From _Matej Veselovac_, Mar 31 2018: (Start)
%C Similarly to the one 3-digit and one 5-digit families given above, at least seven more infinite families exist, for 7-digit consecutive palindromes. Given a nonnegative integer n, we have the following representations palindromic in exactly three consecutive integer number bases j+1, j+2, j+3 :
%C 1. For each j = 36+12n, k = (816 + 2474*j + 3114*j^2 + 2117*j^3 + 852*j^4 + 209*j^5 + 30*j^6 + 2*j^7)/12 is a term of the sequence.
%C 2. For each j = 55+6n, k = (245 + 748 j + 980 j^2 + 718 j^3 + 320 j^4 + 88 j^5 + 14 j^6 + j^7)/6 is a term of the sequence.
%C 3. For each j = 73+2n, k = (247 + 748 j + 980 j^2 + 718 j^3 + 320 j^4 + 88 j^5 + 14 j^6 + j^7)/2 is a term of the sequence.
%C 4. For each j = 116+12n, k = (2440 + 7366 j + 9694 j^2 + 7171 j^3 + 3232 j^4 + 895 j^5 + 142 j^6 + 10 j^7)/12 is a term of the sequence.
%C 5. For each j = 172+6n, k = (812 + 2446 j + 3290 j^2 + 2527 j^3 + 1190 j^4 + 343 j^5 + 56 j^6 + 4 j^7)/6 is a term of the sequence.
%C 6. For each j = 288+12n, k = (1176 + 3566 j + 4374 j^2 + 2807 j^3 + 1032 j^4 + 227 j^5 + 30 j^6 + 2 j^7)/12 is a term of the sequence.
%C 7. For each j = 277+6n, k = (1237 + 3740 j + 4900 j^2 + 3590 j^3 + 1600 j^4 + 440 j^5 + 70 j^6 + 5 j^7)/6 is a term of the sequence.
%C The smallest terms given by these families are of magnitudes ~ 10^10.3, 10^11.5, 10^12.8, 10^14.4, 10^15.5, 10^16.4 and 10^17. The smallest term of the next family, if it exists, is at least of magnitude ~ 10^18.
%C Almost all known terms of the sequence so far belong in one of the above defined families, either being 3-, 5-, or 7- digit palindromes in exactly 3 consecutive integer number bases.
%C There are 13 known terms that do not belong to any families: 300, 3360633, 19987816, 43443858, 532083314, 1778140759, 2721194733, 11325719295, 47622367425, 97638433343, 224678540182, 265282702996, 561091062285 (all but 300 so far are 7-digit cases).
%C Infinite families for consecutive palindromes longer than 7 digits, as well as any examples for those cases, have not yet been observed.
%C Smallest example for 9-digit consecutive palindromes does not exist within first 100 integer number bases, thus is at least > 10^16.
%C Similarly, no terms palindromic in 4 or more consecutive integer number bases have been found, so far.
%C [Extended by _Matej Veselovac_, Feb 05 2019] (End)
%H Matej Veselovac, <a href="/A279093/b279093.txt">Table of n, a(n) for n = 1..10000</a>
%H Matej Veselovac, <a href="/A279093/a279093_2.txt">First seven 7-digit families: digits of palindromes</a>
%e 178 is in the sequence because the bases in which 178 is nontrivially palindromic include 6, 7, and 8: 178 = 454_6 = 343_7 = 262_8.
%e 373 is in the sequence because the bases in which 373 is nontrivially palindromic include 8, 9, and 10: 373 = 565_8 = 454_9 = 373_10.
%e 265282702996 is in the sequence because the bases in which it is nontrivially palindromic include 43, 44, and 45.
%e 130 is nontrivially palindromic in 7 integer bases (11211_3 = 2002_4 = 202_8 = aa_12 = 55_25 = 22_64 = 11_129), but these bases do not include three consecutive integers, so 130 is not in the sequence.
%Y Cf. A002113 (palindromes in base 10), A048268 (smallest palindrome greater than n in bases n and n+1).
%Y Numbers that are palindromic in bases k and k+1: A060792 (k=2), A097928 (k=3), A097929 (k=4), A097930 (k=5), A097931 (k=6), A099145 (k=7), A099146 (k=8), A029965 (k=9), A029966 (k=11).
%Y Cf. A279092 (numbers that are nontrivially palindromic in two or more consecutive integer bases).
%K nonn,base
%O 1,1
%A _Jon E. Schoenfield_, Jan 31 2017