A175064 - OEIS

A175064

a(1) = 1; for n >= 2, a(n) = number of ways h to write the n-th perfect power A001597(n) as m^k with m >= 2 and k >= 1.

1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2

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OFFSET

1,2

COMMENTS

Perfect powers with first occurrence of h >= 2: 4, 16, 64, 65536, 4096, ... [The perfect power corresponding to h is A175065(h) = 2^A005179(h). - Jianing Song, Oct 27 2024]

LINKS

Table of n, a(n) for n=1..86.

FORMULA

a(n) = A000005(A253641(A001597(n))) = A253642(n)+1. - M. F. Hasler, Jan 25 2015

EXAMPLE

For n = 11: A001597(11) = 64; there are 4 ways to write 64 as m^k: 64^1 = 8^2 = 4^3 = 2^6.

PROG

(Python)

from math import gcd

from sympy import mobius, integer_nthroot, divisor_count, factorint

def A175064(n):

if n == 1: return 1