A029956 Numbers that are palindromic in base 11.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 122, 133, 144, 155, 166, 177, 188, 199, 210, 221, 232, 244, 255, 266, 277, 288, 299, 310, 321, 332, 343, 354, 366, 377, 388, 399, 410, 421, 432, 443, 454, 465, 476, 488, 499

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) = 3.4369816... (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, 11], AppendTo[lst, n]], {n, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
pal11Q[n_]:=Module[{idn11=IntegerDigits[n, 11]}, idn11==Reverse[idn11]]; Select[Range[0, 500], pal11Q] (* Harvey P. Dale, May 11 2015 *)
Select[Range[0, 500], PalindromeQ[IntegerDigits[#, 11]] &] (* Michael De Vlieger, May 12 2017, Version 10.3 *)

Prog

(PARI) ispal(n, b)=my(tmp, d=log(n+.5)\log(b)-1); while(d, tmp=n%b; n\=b; if(n\b^d!=tmp, return(0)); n=n%(b^d); d-=2; ); d<0||n%(b+1)==0
is(n)=ispal(n, 11) \\ Charles R Greathouse IV, Aug 21 2012
(PARI) ispal(n, b=11)=my(d=digits(n, b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 04 2020
(Sage)
[n for n in (0..499) if Word(n.digits(11)).is_palindrome()] # Peter Luschny, Sep 13 2018
(Python)
from gmpy2 import digits
from sympy import integer_log
def A029956(n):
    if n == 1: return 0
    y = 11*(x:=11**integer_log(n>>1, 11)[0])
    return int((c:=n-x)*x+int(digits(c, 11)[-2::-1]or'0', 11) if n<x+y else (c:=n-y)*y+int(digits(c, 11)[-1::-1]or'0', 11)) # Chai Wah Wu, Jun 14 2024

Crossrefs

Cf. A002113 (base 10), A029957 (base 12).

Sequence in context: A048308 A043714 A296744 * A297274 A048322 A048335

Adjacent sequences: A029953 A029954 A029955 * A029957 A029958 A029959

Keyword

nonn,base,easy

Author

Patrick De Geest

Status

approved