A380430 Number of powerful numbers k that are not powers of primes (i.e., k is in A286708) that do not exceed the primorial number A002110(n).

0, 0, 0, 0, 7, 50, 254, 1245, 5898, 29600, 163705, 925977, 5690175, 36681963, 241663896, 1662446097, 12134853382, 93406989325, 730785520398, 5990426525483, 50538885715526, 432266550168097, 3845700235189327, 35065304557027821, 334652745159828239, 3262707438761612651

Offset: 0

Author

Michael De Vlieger, Jan 24 2025

Keywords

• nonn
• hard

Examples

			Let P = A002110 and let s = A286708 = A001694 \ A246547 \ {1}.
a(0..3) = 0 since the smallest number in s is 36.
a(4) = 7 since P(4) = 210 and numbers in s that are less than 210 include {36, 72, 100, 108, 144, 196, 200}, etc.
		

Crossrefs

Cf. A001694, A126706, A246547, A286708, A380254, A380402.

Programs

Mathematica

Table[-1 + Sum[If[MoebiusMu[j] != 0, Floor[Sqrt[#/j^3]], 0], {j, #^(1/3)}] - Sum[PrimePi@ Floor[#^(1/k)], {k, 2, Floor[Log2[#]]}] &[Product[Prime[i], {i, n}]], {n, 0, 12} ]

Python

from math import isqrt
from sympy import primorial, primepi, integer_nthroot, mobius
def A380430(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    if n == 0: return 0
    m = primorial(n)
    c, l, j = int(squarefreepi(integer_nthroot(m, 3)[0])-sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))-1), 0, isqrt(m)
    while j>1:
        k2 = integer_nthroot(m//j**2,3)[0]+1
        w = squarefreepi(k2-1)
        c += j*(w-l)
        l, j = w, isqrt(m//k2**3)
    return c-l # Chai Wah Wu, Feb 25 2025

Formula

a(n) = A380254(n) - A380402(n) - 1.

a(n) <= A380403(n) since A286708 is a proper subset of A126706.