A326813 Dirichlet g.f.: zeta(2*s) / (1 - 2^(-s)).

1, 1, 0, 2, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2

Offset

1,4

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Formula

G.f.: Sum_{k>=0} (theta_3(x^(2^k)) - 1) / 2.

a(n) = Sum_{d|n} A209229(n/d) * A010052(d).

a(n) = Sum_{d|n} tau(n/d) * (-1)^bigomega(d) * 2^(omega(2*d) - 1), where tau = A000005, bigomega = A001222 and omega = A001221.

Product_{n>=1} (1 + x^n)^a(n) = g.f. for A001156.

Sum_{k=1..n} a(k) ~ sqrt(2*n) / (sqrt(2)-1). - Vaclav Kotesovec, Oct 20 2019

Multiplicative with a(2^e) = floor(e/2) + 1, and a(p^e) = 0 if e is odd and 1 if e is even, for odd primes p. - Amiram Eldar, Nov 30 2020

Mathematica

Table[Sum[Boole[IntegerQ[Log[2, n/d]]] Boole[IntegerQ[d^(1/2)]], {d, Divisors[n]}], {n, 1, 100}]
Table[Sum[DivisorSigma[0, n/d] (-1)^PrimeOmega[d] 2^(PrimeNu[2 d] - 1), {d, Divisors[n]}], {n, 1, 100}]
f[2, e_] := Floor[e/2] + 1; f[p_, e_] := Boole @ EvenQ[e]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

Crossrefs

Cf. A000005, A001156, A001221, A001222, A001511, A010052, A028983 (positions of 0's), A053866, A209229.

Sequence in context: A088886 A317636 A305566 * A137347 A024941 A219492

Adjacent sequences: A326810 A326811 A326812 * A326814 A326815 A326816

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Oct 19 2019

Status

approved