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

Author

Patrick De Geest

Keywords

• nonn
• base
• easy

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.

Crossrefs

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

Programs

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 *)

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 nChai Wah Wu, Jun 13 2024

Formula

Sum_{n>=2} 1/a(n) = 3.2188878... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020