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A344478 Number of unitary prime divisors p of n such that n/p is squarefree. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
a(p) = 1 for p prime.
LINKS
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
Sequence in context: A356859 A085863 A220354 * A335489 A281188 A323439
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 19 2021
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)