A113850 Numbers whose prime factors are raised to the fifth power.

32, 243, 3125, 7776, 16807, 100000, 161051, 371293, 537824, 759375, 1419857, 2476099, 4084101, 5153632, 6436343, 11881376, 20511149, 24300000, 28629151, 39135393, 45435424, 52521875, 69343957, 79235168, 90224199, 115856201

Offset: 1

Author

Cino Hilliard, Jan 25 2006

Keywords

• easy
• nonn

Examples

			7776 = 32*243 = 2^5*3^5 so the prime factors, 2 and 3, are raised to the fifth power.
		

Crossrefs

Proper subset of A000584.

Nonunit terms of A329332 column 5 in ascending order.

Cf. A005117, A157291.

Programs

Mathematica

Select[ Range@41^5, Union[Last /@ FactorInteger@# ] == {5} &] (* Robert G. Wilson v *)
Rest[Select[Range[100], SquareFreeQ]^5] (* Vaclav Kotesovec, May 22 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 A113850(n):
    def f(x): return int(n+x+1-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**5 # Chai Wah Wu, Sep 13 2024

Formula

Sum_{k>=1} 1/a(k) = zeta(5)/zeta(10) - 1 = A157291 - 1. - Amiram Eldar, May 22 2020

a(n) = A005117(n+1)^5. - Chai Wah Wu, Sep 13 2024

Extensions

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

Offset corrected by Chai Wah Wu, Sep 13 2024