A327810 Numbers that are nontrivially palindromic in two or more consecutive integer bases with non-constant number of digits .

10, 130, 651, 2997, 6643, 6886, 9222, 11950, 26691, 27741, 76449, 175850, 193662, 626626, 704396, 723296, 749470, 755846, 883407, 1181729, 1422773, 1798303, 1817179, 2347506, 2593206, 4252232, 5415589, 10453500, 11435450, 17099841, 17402241, 25651017

Offset: 1

Author

Matej Veselovac, Sep 26 2019

Keywords

• nonn
• base
• hard

Comments

This is a subsequence of A279092. That is, each term of A279092 either has an equal number of digits in all the corresponding consecutive bases, or does not, in which case it belongs to this sequence.

Most numbers in A279092 do not belong to this sequence. That is, "unbalanced" consecutive palindromes are much rarer than "balanced" consecutive palindromes.

Specifically, any subsequence of this sequence that is given by fixing the maximum allowed number of digits in the consecutive bases seems to be finite. In contrast, every such subsequence of A279092 is known to be infinite.

Examples

			Number 10 can be written as 2*4^1+2*4^0=(2,2)_{4} in base four as a palindrome, and as 1*3^2+0*3^1+1*3^0=(1,0,1)_{3} in base three as a palindrome. The bases 4,3 are consecutive, and have 2,3 digits in their representations respectively. All of this makes the number 10=a(1) a term of the sequence.
a(1)  =10    =(2,2)_{4}               =(1,0,1)_{3}
a(2)  =130   =(2,0,0,2)_{4}           =(1,1,2,1,1)_{3}
a(3)  =651   =(3,0,0,3)_{6}           =(1,0,1,0,1)_{5}
a(4)  =2997  =(5,6,6,5)_{8}           =(1,1,5,1,1)_{7}
a(5)  =6643  =(1,0,0,0,1,0,0,0,1)_{3} =(1,1,0,0,1,1,1,1,1,0,0,1,1)_{2}
a(6)  =6886  =(6,8,8,6)_{10}          =(1,0,4,0,1)_{9}
a(7)  =9222  =(2,4,3,3,4,2)_{5}       =(2,1,0,0,0,1,2)_{4}
a(8)  =11950 =(2,3,2,2,2,3,2)_{4}     =(1,2,1,1,0,1,1,2,1)_{3}
a(9)  =26691 =(3,2,3,3,2,3)_{6}       =(1,3,2,3,2,3,1)_{5}
a(10) =27741 =(3,3,2,2,3,3)_{6}       =(1,3,4,1,4,3,1)_{5}
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..74
• Math StackExchange, Finitely many palindromes in two consecutive number bases, for fixed and distinct numbers of digits

Crossrefs

Cf. A279092, A323742.

Programs

Mathematica

c[b1_, d_] := Pick[FromDigits[#, b1 + 1] & /@ #, PalindromeQ[#] && Length[#] > 1 && Length[#] != 2 d + 1 & /@ #] &@IntegerDigits[FromDigits[#, b1] & /@ (Flatten[Outer[List, Range[1, b1 - 1], Sequence @@ ConstantArray[Range[0, b1 - 1], d + 0]], d + 0][[All, Join[Range[d + 1], Reverse[Range[1, d + 0]]]]]), b1 + 1]; a[L_] := DeleteDuplicates[Sort[Select[Join[{10}, Flatten[Table[c[b1, d], {d, 2, Ceiling[Log[2, L]/2] + 1}, {b1, 2, Ceiling[L^(1/(2 d))]}]]], # < L &]]]; a[10^6] (* Matej Veselovac, Sep 28 2019 *)

Extensions

Edited and terms a(11) onward added by Max Alekseyev, Sep 26 2019

Edited by Max Alekseyev, Jun 14 2020