cp's OEIS Frontend

A365491 The number of divisors of the smallest number whose 4th power is divisible by n.

%I A365491 #20 Jul 09 2024 16:52:46
%S A365491 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,
%T A365491 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,
%U A365491 4,8,2,4,2,4,4,4,4,8,2,4,2,4,2,8,4,4,4
%N A365491 The number of divisors of the smallest number whose 4th power is divisible by n.
%C A365491 First differs from A365210 at n = 25 and from A034444 at n = 32.
%C A365491 The number of divisors of the smallest 4th divisible by n, A053167(n), is A365492(n).
%H A365491 Amiram Eldar, <a href="/A365491/b365491.txt">Table of n, a(n) for n = 1..10000</a>
%H A365491 Vaclav Kotesovec, <a href="/A365491/a365491.jpg">Graph - the asymptotic ratio (100000 terms)</a>.
%F A365491 a(n) = A000005(A053166(n)).
%F A365491 Multiplicative with a(p^e) = ceiling(e/4) + 1.
%F A365491 a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).
%F A365491 Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 + 1/p^s - 1/p^(4*s)).
%F A365491 From _Vaclav Kotesovec_, Sep 06 2023: (Start)
%F A365491 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)).
%F A365491 Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(4*s) + 1/p^(5*s)).
%F A365491 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 A365491 f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.57615273538566705952061107826411727540624711680289618854325028459572487...,
%F A365491 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...
%F A365491 and gamma is the Euler-Mascheroni constant A001620. (End)
%F A365491 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
%t A365491 f[p_, e_] := Ceiling[e/4] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%t A365491 With[{c=Range[100]^4},Table[DivisorSigma[0,Surd[SelectFirst[c,Mod[#,n]==0&],4]],{n,90}]] (* _Harvey P. Dale_, Jul 09 2024 *)
%o A365491 (PARI) a(n) = vecprod(apply(x -> (x-1)\4 + 2, factor(n)[, 2]));
%Y A365491 Cf. A000005, A001620, A005117, A053166, A053167, A365492, A365499.
%Y A365491 Cf. A019554, A034444, A322483, A365210.
%K A365491 nonn,easy,mult
%O A365491 1,2
%A A365491 _Amiram Eldar_, Sep 05 2023