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%I #14 Nov 30 2020 03:58:05
%S 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,
%T 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,
%U 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
%N Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s))^2.
%C Dirichlet convolution of A001227 with itself.
%H Amiram Eldar, <a href="/A328486/b328486.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{d|n} A001227(d) * A001227(n/d).
%F 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
%F 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
%p with(numtheory):
%p b:= proc(n) option remember; tau(2*n)-tau(n) end:
%p a:= n-> add(b(d)*b(n/d), d=divisors(n)):
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Oct 16 2019
%t nmax = 80; A001227 = Table[DivisorSum[n, Mod[#, 2] &], {n, 1, nmax}]; Table[DivisorSum[n, A001227[[#]] A001227[[n/#]] &], {n, 1, nmax}]
%t 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 *)
%Y Cf. A000005, A001227, A007426, A070288, A318366, A328487.
%K nonn,mult
%O 1,2
%A _Ilya Gutkovskiy_, Oct 16 2019
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