A014190 Palindromes in base 3 (written in base 10).

0, 1, 2, 4, 8, 10, 13, 16, 20, 23, 26, 28, 40, 52, 56, 68, 80, 82, 91, 100, 112, 121, 130, 142, 151, 160, 164, 173, 182, 194, 203, 212, 224, 233, 242, 244, 280, 316, 328, 364, 400, 412, 448, 484, 488, 524, 560, 572, 608, 644, 656, 692, 728, 730, 757

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• easy

Comments

Rajasekaran, Shallit, & Smith 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
• Patrick De Geest, Palindromic numbers beyond base 10.
• Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
• Aayush Rajasekaran, Jeffrey Shallit, and Tim Smith, Sums of palindromes: an approach via automata, arXiv:1706.10206 [cs.FL], 2017.
• Eric Weisstein's World of Mathematics, Palindromic Number.
• Eric Weisstein's World of Mathematics, Ternary.
• Index entries for sequences that are an additive basis, order 3.

Crossrefs

Cf. A007089, A118594, A134027, A330312 (first differences).

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

Programs

Magma

[n: n in [0..800] | Intseq(n, 3) eq Reverse(Intseq(n, 3))]; // Vincenzo Librandi, Sep 09 2015

Maple

isA014190 := proc(n)
    local L;
    L := convert(n,base,3) ;
    ListTools[Reverse](L) = L ;
end proc:
for n from 0 to 500 do
    if isA014190(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 07 2015

Mathematica

f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,3], AppendTo[lst,n]], {n,1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)

PARI

ispal(n,b=3)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020

Python

from gmpy2 import digits
def A014190(n):
    if n == 1: return 0
    y = 3*(x:=3**(len(digits(n>>1,3))-1))
    return int((c:=n-x)*x+int(digits(c,3)[-2::-1]or'0',3) if nChai Wah Wu, Jun 13 2024

Sage

[n for n in (0..757) if Word(n.digits(3)).is_palindrome()] # Peter Luschny, Sep 13 2018

Formula

Sum_{n>=2} 1/a(n) = 2.61676111... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020