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A344478
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Number of unitary prime divisors p of n such that n/p is squarefree.
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1
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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
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OFFSET
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1,6
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COMMENTS
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a(p) = 1 for p prime.
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LINKS
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FORMULA
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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
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
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MATHEMATICA
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Table[PrimeNu[n] (MoebiusMu[n]^2), {n, 100}] (* Wesley Ivan Hurt, Sep 29 2021 *)
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PROG
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(Python)
from sympy import factorint
fs = factorint(n)
return 0 if len(fs) == 0 or max(fs.values()) > 1 else len(fs) # Chai Wah Wu, Jun 20 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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