A365670 Number of perfect powers k which are not prime powers, and 1 < k < 10^n.

0, 1, 14, 72, 257, 873, 2838, 9085, 28979, 92145, 292832, 930124, 2953569, 9376798, 29760901, 94434276, 299569798, 950072891, 3012393832, 9549260877, 30264906899, 95902117819, 303839485659, 962486295193, 3048497625289, 9654373954803, 30571355398031, 96797106918709

Offset: 1

Author

Peter Luschny, Sep 16 2023

Keywords

• nonn

Comments

k is a perfect power (A001597) <=> there exist integers m and b, b > 1, m >= 1, and k = m^b.

k is a prime power (A246655) <=> there exist integers p and b, b >= 1, p is a prime, and k = p^b.

Examples

			There are 14 perfect powers less than 1000 which are not prime powers:
6^2, 10^2, 12^2, 14^2, 6^3, 15^2, 18^2, 20^2, 21^2, 22^2, 24^2, 26^2, 28^2, 30^2.
		

Links

• Plot2, A267574(n)/A089579(n).

Crossrefs

Cf. A089579, A267574, A131605, A001597, A246655, A024619.

Programs

Python

from sympy import mobius, integer_nthroot, primepi
def A365670(n): return int(sum(mobius(x)*(1-(a:=integer_nthroot(10**n,x)[0]))-primepi(a) for x in range(2,(10**n).bit_length())))-1 if n>1 else n-1 # Chai Wah Wu, Aug 14 2024

SageMath

def A365670(n):
    gen = (p for p in srange(2, 10^n)
           if p.is_perfect_power() and not p.is_prime_power())
    return sum(1 for _ in gen)
print([A365670(n) for n in range(1, 7)])

Formula

a(n) = A089579(n) - A267574(n).

a(n) = card({k: 1 < k < 10^n and k in A131605}).

If k = m^b is a term counted by this sequence then base(k) = m is a term of A024619.

Extensions

Data based on A089579 and A267574.