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

1,5

Comments

Earliest occurrence of k is A018782(k).

Links

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

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A055029 A126812 A008442 * A343221 A327169 A299173

Adjacent sequences: A338687 A338688 A338689 * A338691 A338692 A338693

Keyword

nonn,easy,mult

Author

Jianing Song, Apr 24 2021

Status

approved