A029714 a(n) = Sum_{k divides 3^n} S(k), where S is the Kempner function A002034.

1, 4, 10, 19, 28, 40, 55, 73, 91, 112, 136, 163, 190, 217, 247, 280, 316, 352, 391, 433, 478, 523, 571, 622, 676, 730, 784, 841, 901, 964, 1027, 1093, 1162, 1234, 1306, 1381, 1459, 1540, 1621, 1702, 1783, 1867, 1954, 2044, 2134, 2227, 2323, 2422, 2521, 2623

Offset

0,2

Links

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Formula

a(n) = Sum_{k=0..n-1} A002034(3^k). - Alois P. Heinz, Sep 14 2008

Partial sums of A007844. - Chai Wah Wu, Feb 24 2026

Maple

s:= proc(n) local m; m:= 1; while not type(m!/n, integer) do m:= m+1 od; m end: a:= n-> add(s(3^k), k=0..n-1): seq(a(n), n=1..70); # Alois P. Heinz, Sep 14 2008

Mathematica

S[n_] := S[n] = Module[{k=1}, While[True, If[Divisible[k!, n], Return[k], k++] ] ];
a[n_] := S /@ Divisors[3^n] // Total;
a /@ Range[0, 49] (* Jean-François Alcover, Nov 17 2020 *)

Prog

(Python)
from itertools import count
from sympy import multiplicity
def A029714(n):
    a, m, c = 0, 0, 0
    for k in count(1):
        c += multiplicity(3, k)
        if c+k>=m:
            a+=k*(min(c+k-m, n-m)+1)
            m = c+k+1
        if c+k >= n:
            return 3*a-2 # Chai Wah Wu, Feb 24 2026

Crossrefs

Cf. A029715, A002034, A007844.

Sequence in context: A307395 A380670 A381734 * A348239 A062198 A050858

Adjacent sequences: A029711 A029712 A029713 * A029715 A029716 A029717

Keyword

nonn

Author

Norbert Hungerbuhler (buhler(AT)math.ethz.ch)

Extensions

More terms from Alois P. Heinz, Sep 14 2008

Status

approved