A113851 Numbers whose prime factors are raised to the sixth power.

64, 729, 15625, 46656, 117649, 1000000, 1771561, 4826809, 7529536, 11390625, 24137569, 47045881, 85766121, 113379904, 148035889, 308915776, 594823321, 729000000, 887503681, 1291467969, 1544804416, 1838265625, 2565726409, 3010936384, 3518743761, 4750104241

Offset: 1

Author

Cino Hilliard, Jan 25 2006

Keywords

• easy
• nonn

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..5000

Crossrefs

Subset of A001014. Superset of A030516.

Nonunit terms of A329332 column 6 in ascending order.

Cf. A005117, A269404.

Programs

Maple

for n from 2 to 100 do if(numtheory[issqrfree](n))then printf("%d, ", n^6): fi: od: # Nathaniel Johnston, Jun 21 2011

Mathematica

Select[ Range@37^6, Union[Last /@ FactorInteger@# ] == {6} &] (* Robert G. Wilson v *)
Select[Range[2, 37], SquareFreeQ]^6 (* Amiram Eldar, Oct 13 2020 *)

Python

from math import isqrt
from sympy import mobius
def A113851(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**6 # Chai Wah Wu, Feb 25 2025

Formula

a(n) = A005117(n+1)^6. - Nathaniel Johnston, Jun 21 2011

Sum_{n>=1} 1/a(n) = zeta(6)/zeta(12) - 1 = A269404 - 1. - Amiram Eldar, Oct 13 2020

Extensions

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