A029953 Palindromic in base 6.

0, 1, 2, 3, 4, 5, 7, 14, 21, 28, 35, 37, 43, 49, 55, 61, 67, 74, 80, 86, 92, 98, 104, 111, 117, 123, 129, 135, 141, 148, 154, 160, 166, 172, 178, 185, 191, 197, 203, 209, 215, 217, 259, 301, 343, 385, 427, 434, 476, 518, 560, 602, 644, 651, 693, 735, 777, 819

Offset

1,3

Comments

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 03 2020

Links

T. D. Noe, 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) = 3.03303318... (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, 6], AppendTo[lst, n]], {n, 1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)

Prog

(Magma) [n: n in [0..900] | Intseq(n, 6) eq Reverse(Intseq(n, 6))]; // Vincenzo Librandi, Sep 09 2015
(PARI) ispal(n, b=6)=my(d=digits(n, b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020
(Python)
from gmpy2 import digits
from sympy import integer_log
def A029953(n):
    if n == 1: return 0
    y = 6*(x:=6**integer_log(n>>1, 6)[0])
    return int((c:=n-x)*x+int(digits(c, 6)[-2::-1]or'0', 6) if n<x+y else (c:=n-y)*y+int(digits(c, 6)[-1::-1]or'0', 6)) # Chai Wah Wu, Jun 14 2024

Crossrefs

Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.

Sequence in context: A043709 A296700 A297259 * A048317 A037398 A048331

Adjacent sequences: A029950 A029951 A029952 * A029954 A029955 A029956

Keyword

nonn,base,easy

Author

Patrick De Geest

Status

approved