A029957 - OEIS

%I #30 Jun 14 2024 11:59:19

%S 0,1,2,3,4,5,6,7,8,9,10,11,13,26,39,52,65,78,91,104,117,130,143,145,

%T 157,169,181,193,205,217,229,241,253,265,277,290,302,314,326,338,350,

%U 362,374,386,398,410,422,435,447,459,471,483,495,507,519,531

%N Numbers that are palindromic in base 12.

%C Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - _Charles R Greathouse IV_, May 04 2020

%H John Cerkan, <a href="/A029957/b029957.txt">Table of n, a(n) for n = 1..10000</a>

%H Javier Cilleruelo, Florian Luca and Lewis Baxter, <a href="https://doi.org/10.1090/mcom/3221">Every positive integer is a sum of three palindromes</a>, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, <a href="http://arxiv.org/abs/1602.06208">arXiv preprint</a>, arXiv:1602.06208 [math.NT], 2017.

%H Patrick De Geest, <a href="http://www.worldofnumbers.com/nobase10.htm">Palindromic numbers beyond base 10</a>.

%H Phakhinkon Phunphayap and Prapanpong Pongsriiam, <a href="https://doi.org/10.13140/RG.2.2.23202.79047">Estimates for the Reciprocal Sum of b-adic Palindromes</a>, 2019.

%H <a href="/index/Ab#basis_03">Index entries for sequences that are an additive basis</a>, order 3.

%F Sum_{n>=2} 1/a(n) = 3.4989489... (Phunphayap and Pongsriiam, 2019). - _Amiram Eldar_, Oct 17 2020

%t f[n_,b_]:=Module[{i=IntegerDigits[n,b]},i==Reverse[i]];lst={};Do[If[f[n,12],AppendTo[lst,n]],{n,7!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jul 08 2009 *)

%o (PARI) isok(n) = my(d=digits(n, 12)); d == Vecrev(d); \\ _Michel Marcus_, May 13 2017

%o (Python)

%o from sympy import integer_log

%o from gmpy2 import digits

%o def A029957(n):

%o     if n == 1: return 0

%o     y = 12*(x:=12**integer_log(n>>1,12)[0])

%o     return int((c:=n-x)*x+int(digits(c,12)[-2::-1]or'0',12) if n<x+y else (c:=n-y)*y+int(digits(c,12)[-1::-1]or'0',12)) # _Chai Wah Wu_, Jun 14 2024

%Y Cf. A029958, A029959, A029960 (in bases 13..15).

%K nonn,base,easy

%O 1,3

%A _Patrick De Geest_