A029959 Numbers that are palindromic in base 14.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 197, 211, 225, 239, 253, 267, 281, 295, 309, 323, 337, 351, 365, 379, 394, 408, 422, 436, 450, 464, 478, 492, 506, 520, 534, 548, 562, 576, 591

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) = 3.6112482... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

Example

195 is DD in base 14.
196 is 100 in base 14, so it's not in the sequence.
197 is 101 in base 14.

Mathematica

palQ[n_, b_:10] := Module[{idn = IntegerDigits[n, b]}, idn == Reverse[idn]]; Select[ Range[0, 600], palQ[#, 14] &] (* Harvey P. Dale, Aug 03 2014 *)

Prog

(PARI) isok(n) = Pol(d=digits(n, 14)) == Polrev(d); \\ Michel Marcus, Mar 12 2017
(Python)
from sympy import integer_log
from gmpy2 import digits
def A029959(n):
    if n == 1: return 0
    y = 14*(x:=14**integer_log(n>>1, 14)[0])
    return int((c:=n-x)*x+int(digits(c, 14)[-2::-1]or'0', 14) if n<x+y else (c:=n-y)*y+int(digits(c, 14)[-1::-1]or'0', 14)) # Chai Wah Wu, Jun 14 2024

Crossrefs

Palindromes in bases 2 through 13: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957, A029958.

Sequence in context: A048311 A043717 A296753 * A297283 A048325 A048338

Adjacent sequences: A029956 A029957 A029958 * A029960 A029961 A029962

Keyword

nonn,base,easy

Author

Patrick De Geest

Status

approved