A344478 Number of unitary prime divisors p of n such that n/p is squarefree.

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 0, 1, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 3, 1, 0, 2, 2, 2, 0, 1, 2, 2, 0, 1, 3, 1, 0, 0, 2, 1, 0, 0, 0, 2, 0, 1, 0, 2, 0, 2, 2, 1, 0, 1, 2, 0, 0, 2, 3, 1, 0, 2, 3, 1, 0, 1, 2, 0, 0, 2, 3, 1, 0, 0, 2, 1, 0, 2, 2, 2, 0, 1, 0, 2, 0, 2, 2

Offset

1,6

Comments

a(p) = 1 for p prime.

Links

Table of n, a(n) for n=1..94.

Tanay Wakhare, Sums involving the number of distinct prime factors function, arXiv:1604.05671 [math.HO], 2016-2017.

Formula

a(n) = Sum_{p|n, gcd(p,n/p) = 1} mu(n/p)^2, where mu is the Möbius function (A008683).

For n > 1, a(n) = 0 if n is not squarefree. a(n) = omega(n) (A001221) if n is squarefree. - Chai Wah Wu, Jun 20 2021

a(n) = omega(n) * mu(n)^2. - Wesley Ivan Hurt, Sep 29 2021

Dirichlet g.f.: (zeta(s)/zeta(2*s)) * P(s, 1), where P(s, c) = Sum_{p prime} 1/(p^s + c) is the shifted prime zeta function (Wakhare, 2016). - Amiram Eldar, Sep 19 2023

Mathematica

Table[PrimeNu[n] (MoebiusMu[n]^2), {n, 100}] (* Wesley Ivan Hurt, Sep 29 2021 *)

Prog

(Python)
from sympy import factorint
def A344478(n):
    fs = factorint(n)
    return 0 if len(fs) == 0 or max(fs.values()) > 1 else len(fs) # Chai Wah Wu, Jun 20 2021

Crossrefs

Cf. A001221, A008683.

Sequence in context: A356859 A085863 A220354 * A335489 A281188 A323439

Adjacent sequences: A344475 A344476 A344477 * A344479 A344480 A344481

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 19 2021

Status

approved