A361264 - OEIS

%I #13 Sep 01 2023 04:09:28

%S 1,8,27,16,125,216,343,32,81,1000,1331,432,2197,2744,3375,64,4913,648,

%T 6859,2000,9261,10648,12167,864,625,17576,243,5488,24389,27000,29791,

%U 128,35937,39304,42875,1296,50653,54872,59319,4000,68921,74088,79507,21296,10125

%N Multiplicative with a(p^e) = p^(e + 2), e > 0.

%H Amiram Eldar, <a href="/A361264/b361264.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F Sum_{k=1..n} a(k) ~ c * zeta(3) * n^4 / 4, where c = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 = 0.53589615382833799980850263131854595064822237...

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

%F a(n) = n * A007947(n)^2 = A064549(n) * A007947(n) = A064549(A064549(n)).

%F A000005(a(n)) = A360997(n).

%F Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2*(p-1))) = A065483. (End)

%t g[p_, e_] := p^(e+2); a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, 1 + p^3*X/(1 - p*X))[n], ", "))

%Y Cf. A000005, A003557, A064549, A065483, A007947, A330523, A360997, A361266.

%K nonn,easy,mult

%O 1,2

%A _Vaclav Kotesovec_, Mar 06 2023