A029803 Numbers that are palindromic in base 8.

0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 27, 36, 45, 54, 63, 65, 73, 81, 89, 97, 105, 113, 121, 130, 138, 146, 154, 162, 170, 178, 186, 195, 203, 211, 219, 227, 235, 243, 251, 260, 268, 276, 284, 292, 300, 308, 316, 325, 333, 341, 349, 357, 365, 373, 381, 390, 398

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) = 3.2188878... (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, 8], AppendTo[lst, n]], {n, 1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)

Prog

(PARI) ispal(n, b=8)=my(d=digits(n, b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020
(Python)
from itertools import chain, count, islice
def A029803_gen(): # generator of terms
    return chain((0, ), chain.from_iterable(chain((int((s:=oct(d)[2:])+s[-2::-1], 8) for d in range(8**l, 8**(l+1))), (int((s:=oct(d)[2:])+s[::-1], 8) for d in range(8**l, 8**(l+1)))) for l in count(0)))
A029803_list = list(islice(A029803_gen(), 20)) # Chai Wah Wu, Jun 23 2022
(Python)
def A029803(n):
    if n == 1: return 0
    y = (x:=1<<(m:=n.bit_length()-2)-m%3)<<3
    return (c:=n-x)*x+int(oct(c)[-2:1:-1]or'0', 8) if n<x+y else (c:=n-y)*y+int(oct(c)[-1:1:-1]or'0', 8) # Chai Wah Wu, Jun 13 2024

Crossrefs

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

Sequence in context: A048305 A043711 A296706 * A297265 A048319 A037405

Adjacent sequences: A029800 A029801 A029802 * A029804 A029805 A029806

Keyword

nonn,base,easy

Author

Patrick De Geest

Status

approved