cp's OEIS Frontend

A380922 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s + 1/p^(3*s)).

%I A380922 #41 Apr 22 2025 14:17:51
%S A380922 1,2,2,2,2,4,2,3,2,4,2,4,2,4,4,3,2,4,2,4,4,4,2,6,2,4,3,4,2,8,2,3,4,4,
%T A380922 4,4,2,4,4,6,2,8,2,4,4,4,2,6,2,4,4,4,2,6,4,6,4,4,2,8,2,4,4,3,4,8,2,4,
%U A380922 4,8,2,6,2,4,4,4,4,8,2,6,3,4,2,8,4,4,4,6,2,8,4,4,4,4,4,6,2,4,4,4
%N A380922 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s + 1/p^(3*s)).
%C A380922 First differs from A061389 at n = 32.
%C A380922 First differs from A322483 at n = 32.
%C A380922 First differs from A372380 at n = 128 (next differences are at n=128*k, n=2187*k, ...).
%C A380922 The number of divisors of n that are both biquadratefree (A046100) and exponentially odd (A268335), i.e., in A336591. - _Amiram Eldar_, Apr 22 2025
%H A380922 Vaclav Kotesovec, <a href="/A380922/b380922.txt">Table of n, a(n) for n = 1..10000</a>
%H A380922 Vaclav Kotesovec, <a href="/A380922/a380922.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%F A380922 Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 1/p^(4*s)).
%F A380922 Dirichlet g.f.: zeta(s)^2 * f(s).
%F A380922 Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
%F A380922 f(1) = Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.684286924186862318141968725791218083472312736723163777284618226290055...,
%F A380922 f'(1) = f(1) * Sum_{p prime} (2*p^2 - 3*p + 4) * log(p) / ((p-1) * (p^3 + p^2 + 1)) = f(1) * 0.85825768698295295413525347933038488513032293516964600096226328323449...
%F A380922 and gamma is the Euler-Mascheroni constant A001620.
%F A380922 Multiplicative with a(p^e) = 2 if e <= 2 and 3 otherwise. - _Amiram Eldar_, Apr 22 2025
%t A380922 f[p_, e_] := If[e < 3, 2, 3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Apr 22 2025 *)
%o A380922 (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * (1 + X + X^3))[n], ", "))
%Y A380922 Cf. A073184, A365498, A383292.
%Y A380922 Cf. A322483, A372380, A061389.
%Y A380922 Cf. A046100, A268335, A336591.
%K A380922 nonn,mult,easy
%O A380922 1,2
%A A380922 _Vaclav Kotesovec_, Apr 22 2025