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A365491
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The number of divisors of the smallest number whose 4th power is divisible by n.
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4
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1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 3, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 3, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4
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OFFSET
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1,2
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COMMENTS
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The number of divisors of the smallest 4th divisible by n, A053167(n), is A365492(n).
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LINKS
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FORMULA
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Multiplicative with a(p^e) = ceiling(e/4) + 1.
a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 + 1/p^s - 1/p^(4*s)).
Dirichlet g.f.: zeta(s)^2 * zeta(4*s) * Product_{p prime} (1 - 1/p^(2*s) - 1/p^(4*s) + 1/p^(5*s)).
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(4*s) + 1/p^(5*s)).
Sum_{k=1..n} a(k) ~ zeta(4) * f(1) * n * (log(n) + 2*gamma - 1 + 4*zeta'(4)/zeta(4) + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.57615273538566705952061107826411727540624711680289618854325028459572487...,
f'(1) = f(1) * Sum_{p prime} (-5 + 4*p + 2*p^3) * log(p) / (1 - p - p^3 + p^5) = f(1) * 1.3011434396559802378314782600747661399223385669839998680418996210...
and gamma is the Euler-Mascheroni constant A001620. (End)
a(n) = A322483(A019554(n)) (the number of exponentially odd divisors of the smallest number whose square is divisible by n). - Amiram Eldar, Sep 08 2023
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MATHEMATICA
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f[p_, e_] := Ceiling[e/4] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
With[{c=Range[100]^4}, Table[DivisorSigma[0, Surd[SelectFirst[c, Mod[#, n]==0&], 4]], {n, 90}]] (* Harvey P. Dale, Jul 09 2024 *)
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PROG
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(PARI) a(n) = vecprod(apply(x -> (x-1)\4 + 2, factor(n)[, 2]));
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CROSSREFS
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KEYWORD
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nonn,easy,mult,changed
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AUTHOR
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STATUS
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approved
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