A326813 - OEIS

%I #18 Nov 30 2020 03:58:15

%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,

%T 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,

%U 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

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

%H Amiram Eldar, <a href="/A326813/b326813.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A326813/a326813_1.jpg">Graph - the asymptotic ratio (1000000 terms)</a>

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

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

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

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

%F Sum_{k=1..n} a(k) ~ sqrt(2*n) / (sqrt(2)-1). - _Vaclav Kotesovec_, Oct 20 2019

%F 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

%t Table[Sum[Boole[IntegerQ[Log[2, n/d]]] Boole[IntegerQ[d^(1/2)]], {d, Divisors[n]}], {n, 1, 100}]

%t Table[Sum[DivisorSigma[0, n/d] (-1)^PrimeOmega[d] 2^(PrimeNu[2 d] - 1), {d, Divisors[n]}], {n, 1, 100}]

%t 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 *)

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

%K nonn,mult

%O 1,4

%A _Ilya Gutkovskiy_, Oct 19 2019