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

1,1

Links

Table of n, a(n) for n=1..26.

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

Example

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

Mathematica

Select[ Range@41^5, Union[Last /@ FactorInteger@# ] == {5} &] (* Robert G. Wilson v *)
Rest[Select[Range[100], SquareFreeQ]^5] (* Vaclav Kotesovec, May 22 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 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

Crossrefs

Proper subset of A000584.

Nonunit terms of A329332 column 5 in ascending order.

Cf. A005117, A157291.

Sequence in context: A343325 A343285 A153159 * A046454 A050997 A056572

Adjacent sequences: A113847 A113848 A113849 * A113851 A113852 A113853

Keyword

easy,nonn

Author

Cino Hilliard, Jan 25 2006

Extensions

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

Offset corrected by Chai Wah Wu, Sep 13 2024

Status

approved