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A365549
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The number of exponentially odd divisors of the square root of the largest square dividing n.
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4
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1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1
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OFFSET
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1,4
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COMMENTS
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The number of exponentially odd divisors of the largest square dividing n is the same as the number of squares dividing n, A046951(n).
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LINKS
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FORMULA
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a(n) >= 1 with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = 2 + floor((e-2)/4).
Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 + 1/p^(2*s) - 1/p^(4*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4) = 1.54211628314015874165... .
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MATHEMATICA
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f[p_, e_] := 2 + Floor[(e-2)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PROG
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(PARI) a(n) = vecprod(apply(x -> 2 + (x-2)\4, factor(n)[, 2]));
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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