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A351394
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Number of divisors of n that are either squarefree, prime powers, or both.
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1
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1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 5, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2, 8, 2, 6, 4, 4, 4, 6, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 3, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 9, 2, 4, 5, 7, 4, 8, 2, 5, 4, 8, 2, 7, 2, 4, 5, 5, 4, 8, 2, 7, 5, 4, 2, 9, 4, 4, 4, 6, 2, 9, 4, 5, 4, 4, 4
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = Sum_{d|n} sign(mu(d)^2 + [omega(d) = 1]).
a(n) = Sum_{d|n} (mu(d)^2 + [omega(d) = 1]*(1 - mu(d)^2)).
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EXAMPLE
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a(36) = 6; 36 has 4 squarefree divisors 1,2,3,6 (where the primes 2 and 3 are both squarefree and 1st powers of primes) and 2 (additional) divisors that are powers of primes, 2^2 and 3^2.
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MATHEMATICA
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a[n_] := Module[{e = FactorInteger[n][[;; , 2]], nu, omega}, nu = Length[e]; omega = Total[e]; 2^nu + omega - nu]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Oct 06 2023 *)
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PROG
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(PARI) a(n) = {my(f = factor(n), nu = omega(f), om = bigomega(f)); 2^nu + om - 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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