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A365492 The number of divisors of the smallest 4th power divisible by n.

%I A365492 #7 Sep 06 2023 01:16:43
%S A365492 1,5,5,5,5,25,5,5,5,25,5,25,5,25,25,5,5,25,5,25,25,25,5,25,5,25,5,25,
%T A365492 5,125,5,9,25,25,25,25,5,25,25,25,5,125,5,25,25,25,5,25,5,25,25,25,5,
%U A365492 25,25,25,25,25,5,125,5,25,25,9,25,125,5,25,25,125,5,25
%N A365492 The number of divisors of the smallest 4th power divisible by n.
%C A365492 First differs from A082476 at n = 32.
%C A365492 The number of divisors of the 4th root of the smallest 4th power divisible by n, A053166(n), is A365491(n).
%H A365492 Amiram Eldar, <a href="/A365492/b365492.txt">Table of n, a(n) for n = 1..10000</a>
%F A365492 a(n) = A000005(A053167(n)).
%F A365492 Multiplicative with a(p^e) = 4*ceiling(e/4) + 1.
%F A365492 Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 + 4/p^s - 1/p^(4*s)).
%t A365492 f[p_, e_] := 4*Ceiling[e/4] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365492 (PARI) a(n) = vecprod(apply(x -> 4*((x-1)\4) + 5, factor(n)[, 2]));
%Y A365492 Cf. A000005, A053166, A053167, A082476, A365491.
%K A365492 nonn,easy,mult
%O A365492 1,2
%A A365492 _Amiram Eldar_, Sep 05 2023