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A351414
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Number of divisors of n that are either prime or have at least 1 square divisor > 1 and at least two distinct prime factors.
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2
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0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 3, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 1, 2, 3, 1, 3, 2, 3, 1, 7, 1, 2, 3, 3, 2, 3, 1, 5, 1, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 2, 2, 2, 6
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OFFSET
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1,6
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LINKS
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FORMULA
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a(n) = Sum_{d|n} [[omega(d) = 1] = mu(d)^2], where [ ] is the Iverson bracket.
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EXAMPLE
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a(24) = 4; 24 has divisors 2,3 (primes) and 12,24 (which both have at least 1 square divisor > 1 and at least two distinct prime factors).
a(36) = 5; 36 has divisors 2,3 (primes) and 12,18,36 (which all have at least 1 square divisor > 1 and at least two distinct prime factors).
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MATHEMATICA
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a[n_] := Module[{e = FactorInteger[n][[;; , 2]], d, nu, omega}, d = Times @@ (e+1); nu = Length[e]; omega = Total[e]; d - 2^nu - omega + 2*nu]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Oct 06 2023 *)
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PROG
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(PARI) a(n) = {my(f = factor(n), d = numdiv(f), nu = omega(f), om = bigomega(f)); d - 2^nu - om + 2*nu; } \\ Amiram Eldar, Oct 06 2023
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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