A029954 Palindromic in base 7.

0, 1, 2, 3, 4, 5, 6, 8, 16, 24, 32, 40, 48, 50, 57, 64, 71, 78, 85, 92, 100, 107, 114, 121, 128, 135, 142, 150, 157, 164, 171, 178, 185, 192, 200, 207, 214, 221, 228, 235, 242, 250, 257, 264, 271, 278, 285, 292, 300, 307, 314, 321, 328, 335, 342, 344, 400, 456

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,7], AppendTo[lst,n]], {n,1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
pal7Q[n_]:=Module[{idn7=IntegerDigits[n,7]},idn7==Reverse[idn7]]; Select[ Range[0,500],pal7Q] (* Harvey P. Dale, Jul 30 2015 *)

PARI

ispal(n,b=7)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020

Python

from gmpy2 import digits
def palQgen(l,b): # generator of palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,l+1):
            for y in range(b**(x-1),b**x):
                s = digits(y,b)
                yield int(s+s[-2::-1],b)
            for y in range(b**(x-1),b**x):
                s = digits(y,b)
                yield int(s+s[::-1],b)
A029954_list = list(palQgen(4,7)) # Chai Wah Wu, Dec 01 2014

Python

from gmpy2 import digits
from sympy import integer_log
def A029954(n):
    if n == 1: return 0
    y = 7*(x:=7**integer_log(n>>1,7)[0])
    return int((c:=n-x)*x+int(digits(c,7)[-2::-1]or'0',7) if nChai Wah Wu, Jun 14 2024

Formula

Sum_{n>=2} 1/a(n) = 3.1313768... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020