A338690 - OEIS

%I #21 Oct 22 2022 08:06:29

%S 1,0,0,1,2,0,0,0,1,0,0,0,2,0,0,1,2,0,0,2,0,0,0,0,3,0,0,0,2,0,0,0,0,0,

%T 0,1,2,0,0,0,2,0,0,0,2,0,0,0,1,0,0,2,2,0,0,0,0,0,0,0,2,0,0,1,4,0,0,2,

%U 0,0,0,0,2,0,0,0,0,0,0,2,1,0,0,0,4,0,0,0,2,0,0,0,0,0,0,0,2,0,0,3

%N Inverse Moebius transform of A209615.

%C Earliest occurrence of k is A018782(k).

%H Jianing Song, <a href="/A338690/b338690.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p = 2 or p == 3 (mod 4).

%F a(n) = A002654(n) = A035184(n) for odd n. a(2^e * m) = a(m) for even m, 0 for odd m.

%F Dirichlet g.f.: zeta(s)*beta(s)/(1 + 2^(-s)).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/6 = 0.523598... (A019673). - _Amiram Eldar_, Oct 22 2022

%t f[p_, e_] := If[Mod[p, 4] == 1, e + 1, (1 + (-1)^e)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 22 2022 *)

%o (PARI) a(n) = my(r=1, f=factor(n)); for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); if(p%4==1, r*=e+1, if(e%2, return(0)))); r

%Y Cf. A209615, A035184 (a similar sequence), A018782, A002654, A019673.

%K nonn,easy,mult

%O 1,5

%A _Jianing Song_, Apr 24 2021