A365498 - OEIS

%I #20 Sep 09 2023 05:38:12

%S 1,2,2,2,2,4,2,1,2,4,2,4,2,4,4,1,2,4,2,4,4,4,2,2,2,4,1,4,2,8,2,1,4,4,

%T 4,4,2,4,4,2,2,8,2,4,4,4,2,2,2,4,4,4,2,2,4,2,4,4,2,8,2,4,4,1,4,8,2,4,

%U 4,8,2,2,2,4,4,4,4,8,2,2,1,4,2,8,4,4,4

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

%C The number of unitary divisors of n that are cubefree numbers (A004709). - _Amiram Eldar_, Sep 06 2023

%H Vaclav Kotesovec, <a href="/A365498/b365498.txt">Table of n, a(n) for n = 1..10000</a>

%F Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).

%F Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).

%F Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where

%F f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085026313185459506482223745141452711510108346133288...,

%F f'(1) = f(1) * Sum_{p prime} (-4 + 3*p + 2*p^2) * log(p) / (1 - p - p^2 + p^4) = f(1) * 1.4525924794451595590371439593828547341482465114411929136723476679...

%F and gamma is the Euler-Mascheroni constant A001620.

%F Multiplicative with a(p^e) = 2 if e <= 2, and 1 otherwise. - _Amiram Eldar_, Sep 06 2023

%t f[p_, e_] := If[e <= 2, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 06 2023 *)

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * (1 + X - X^3))[n], ", "))

%Y Cf. A001620, A004709, A056671, A365488, A365499.

%K nonn,easy,mult

%O 1,2

%A _Vaclav Kotesovec_, Sep 06 2023