A113852 Numbers whose prime factors are raised to the seventh power.

128, 2187, 78125, 279936, 823543, 10000000, 19487171, 62748517, 105413504, 170859375, 410338673, 893871739, 1801088541, 2494357888, 3404825447, 8031810176, 17249876309, 21870000000, 27512614111, 42618442977, 52523350144, 64339296875, 94931877133, 114415582592

Offset: 1

Author

Cino Hilliard, Jan 25 2006

Keywords

• easy
• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Proper subset of A001015.

Nonunit terms of A329332 column 7 in ascending order.

Cf. A005117, A013665, A013672.

Programs

Mathematica

Select[Range@34^7, Union[Last /@ FactorInteger@# ] == {7} &] (* Robert G. Wilson v, Jan 26 2006 *)
Select[Range[2, 34], SquareFreeQ]^7 (* Amiram Eldar, Oct 13 2020 *)

PARI

allpwrfact(n,p) = /* All prime factors are raised to the power p */ { local(x,j,ln,y,flag); for(x=4,n, y=Vec(factor(x)); ln = length(y[1]); flag=0; for(j=1,ln, if(y[2][j]==p,flag++); ); if(flag==ln,print1(x",")); ) }

Python

from math import isqrt
from sympy import mobius
def A113852(n):
    def f(x): return int(n+1-sum(mobius(k)*(x//k**2) for k in range(2, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**7 # Chai Wah Wu, Feb 25 2025

Formula

From Amiram Eldar, Oct 13 2020: (Start)

a(n) = A005117(n+1)^7.

Sum_{n>=1} 1/a(n) = zeta(7)/zeta(14) - 1. (End)

Extensions

More terms from Robert G. Wilson v, Jan 26 2006