A029957 Numbers that are palindromic in base 12.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 145, 157, 169, 181, 193, 205, 217, 229, 241, 253, 265, 277, 290, 302, 314, 326, 338, 350, 362, 374, 386, 398, 410, 422, 435, 447, 459, 471, 483, 495, 507, 519, 531

Offset

1,3

Comments

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

Links

John Cerkan, Table of n, a(n) for n = 1..10000

Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.

Patrick De Geest, Palindromic numbers beyond base 10.

Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.

Index entries for sequences that are an additive basis, order 3.

Formula

Sum_{n>=2} 1/a(n) = 3.4989489... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

Mathematica

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 *)

Prog

(PARI) isok(n) = my(d=digits(n, 12)); d == Vecrev(d); \\ Michel Marcus, May 13 2017
(Python)
from sympy import integer_log
from gmpy2 import digits
def A029957(n):
    if n == 1: return 0
    y = 12*(x:=12**integer_log(n>>1, 12)[0])
    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

Crossrefs

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

Sequence in context: A048309 A043715 A296747 * A297277 A048323 A048336

Adjacent sequences: A029954 A029955 A029956 * A029958 A029959 A029960

Keyword

nonn,base,easy

Author

Patrick De Geest

Status

approved