%I #14 Sep 01 2023 04:09:05

%S 1,4,4,5,4,16,4,6,5,16,4,20,4,16,16,7,4,20,4,20,16,16,4,24,5,16,6,20,

%T 4,64,4,8,16,16,16,25,4,16,16,24,4,64,4,20,20,16,4,28,5,20,16,20,4,24,

%U 16,24,16,16,4,80,4,16,20,9,16,64,4,20,16,64,4,30,4,16

%N Multiplicative with a(p^e) = e + 3.

%H Amiram Eldar, <a href="/A360997/b360997.txt">Table of n, a(n) for n = 1..10000</a>

%F Dirichlet g.f.: Product_{primes p} (1 + (4*p^s - 3)/(p^s - 1)^2).

%F Dirichlet g.f.: zeta(s)^4 * Product_{primes p} (1 - 5/p^(2*s) + 6/p^(3*s) - 2/p^(4*s)).

%F From _Amiram Eldar_, Sep 01 2023: (Start)

%F a(n) = A000005(A361264(n)).

%F a(n) = A074816(n)*A007426(n)/A007425(n). (End)

%t g[p_, e_] := e+3; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1+2*X-2*X^2)/(1-X)^2)[n], ", "))

%Y Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), this sequence (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), A360909 (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).

%Y Cf. A000005, A007425, A007426, A074816.

%K nonn,easy,mult

%O 1,2

%A _Vaclav Kotesovec_, Feb 28 2023