A380403 Number of integers k that are neither squarefree nor prime powers (in A126706) and that do not exceed primorial A002110(n).

0, 0, 0, 5, 67, 871, 11693, 199976, 3802411, 87466676, 2536583089, 78634293907, 2909470106300, 119288281458176, 5129396144497507, 241081619059363357, 12777325812023481231, 753862222923258499554

Offset: 0

Author

Michael De Vlieger, Jan 23 2025

Keywords

• nonn
• hard
• more

Examples

			Let s = A126706 and let P(n) = A002110(n).
a(0..2) = 0 since P(0..2) = {1, 2, 6}, and the smallest number in s is 12.
a(3) = 5 since P(3) = 30, and the set s(1..6) = {12, 18, 20, 24, 28} contains k <= 30.
a(4) = 67 since P(4) = 210, and the set s(1..67) = {12, 18, 20, ..., 207, 208} contains k <= 210, etc.
		

Crossrefs

Cf. A002110, A126706, A158341, A380402.

Programs

Mathematica

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

PARI

a(n) = my(q=vecprod(primes(n))); q - sum(k=2, logint(q, 2), primepi(sqrtnint(q, k))) - sum(k=1, sqrtint(q), q\k^2*moebius(k)); \\ Jinyuan Wang, Feb 25 2025

Python

from math import isqrt
from sympy import primorial, primepi, integer_nthroot, mobius
def A380403(n):
    if n == 0: return 0
    m = primorial(n)
    return int(-sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))-sum(mobius(k)*(m//k**2) for k in range(2, isqrt(m)+1))) # Chai Wah Wu, Jan 24 2025

Formula

a(n) = P(n) - (Sum_{k=2..floor(log_2(P(n)))} pi(floor(P(n)^(1/k)))) - Sum_{k=1..floor(sqrt(P(n)))} mu(k)*floor(P(n)/(k^2)), where P(n) = A002110(n).

a(n) = A002110(n) - A380402(n) - A158341(n) - 1.

Extensions

Offset changed to 0 by Jinyuan Wang, Jan 24 2025

a(16) from Chai Wah Wu, Jan 24 2025

a(17) from Chai Wah Wu, Jan 25 2025