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

1,3

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) = 3.1313768... (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, 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 *)

Prog

(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 n<x+y else (c:=n-y)*y+int(digits(c, 7)[-1::-1]or'0', 7)) # Chai Wah Wu, Jun 14 2024
(PARI) ispal(n, b=7)=my(d=digits(n, b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020

Crossrefs

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

Sequence in context: A043710 A296703 A297262 * A048318 A037402 A048332

Adjacent sequences: A029951 A029952 A029953 * A029955 A029956 A029957

Keyword

nonn,base,easy,changed

Author

Patrick De Geest

Status

approved