A338690 Inverse Moebius transform of A209615.

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, 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, 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

Offset: 1

Author

Jianing Song, Apr 24 2021

Keywords

• nonn
• easy
• mult

Comments

Earliest occurrence of k is A018782(k).

Links

• Jianing Song, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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 *)

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

Formula

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).

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

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

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