A328486 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s))^2.

1, 2, 4, 3, 4, 8, 4, 4, 10, 8, 4, 12, 4, 8, 16, 5, 4, 20, 4, 12, 16, 8, 4, 16, 10, 8, 20, 12, 4, 32, 4, 6, 16, 8, 16, 30, 4, 8, 16, 16, 4, 32, 4, 12, 40, 8, 4, 20, 10, 20, 16, 12, 4, 40, 16, 16, 16, 8, 4, 48, 4, 8, 40, 7, 16, 32, 4, 12, 16, 32, 4, 40, 4, 8, 40, 12, 16, 32, 4, 20

Offset

1,2

Comments

Dirichlet convolution of A001227 with itself.

Links

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

Formula

a(n) = Sum_{d|n} A001227(d) * A001227(n/d).

Sum_{k=1..n} a(k) ~ n * (log(n)^3/24 + (g/2 + log(2)/4 - 1/8)* log(n)^2 + (1/4 - g + 3*g^2/2 - log(2)/2 + 2*g*log(2) - sg1)* log(n) - 1/4 + (1 - 2*log(2))*g + (3*log(2) - 3/2)*g^2 + g^3 + log(2)/2 - log(2)^3/6 + (1 - 3*g - 2*log(2))* sg1 + sg2/2), where g is the Euler-Mascheroni constant A001620 and sg1, sg2 are the Stieltjes constants, see A082633 and A086279. - Vaclav Kotesovec, Oct 17 2019

Multiplicative with a(2^e) = e + 1, and a(p^e) = (e + 1)*(e + 2)*(e + 3)/6 for odd primes p. - Amiram Eldar, Nov 30 2020

Maple

with(numtheory):
b:= proc(n) option remember; tau(2*n)-tau(n) end:
a:= n-> add(b(d)*b(n/d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 16 2019

Mathematica

nmax = 80; A001227 = Table[DivisorSum[n, Mod[#, 2] &], {n, 1, nmax}]; Table[DivisorSum[n, A001227[[#]] A001227[[n/#]] &], {n, 1, nmax}]
f[2, e_] := e + 1; f[p_, e_] := (e + 1)*(e + 2)*(e + 3)/6; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

Crossrefs

Cf. A000005, A001227, A007426, A070288, A318366, A328487.

Sequence in context: A229047 A335841 A133702 * A349606 A332224 A080001

Adjacent sequences: A328483 A328484 A328485 * A328487 A328488 A328489

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Oct 16 2019

Status

approved