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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

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)

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 *)

PROG

(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

CROSSREFS

Proper subset of A001015.

Nonunit terms of A329332 column 7 in ascending order.

Cf. A005117, A013665, A013672.