A361268 - OEIS

%I #20 Sep 01 2023 02:55:22

%S 1,8,27,32,125,216,343,96,162,1000,1331,864,2197,2744,3375,256,4913,

%T 1296,6859,4000,9261,10648,12167,2592,1250,17576,729,10976,24389,

%U 27000,29791,640,35937,39304,42875,5184,50653,54872,59319,12000,68921,74088,79507,42592

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

%C In general, if the function is multiplicative with a(p^e) = e*p^(e+m) where m > 0, then Dirichlet g.f.: Product_{primes p} (1 + p^(s + m + 1)/(p^s - p)^2).

%C Equivalently, Dirichlet g.f.: zeta(s-m-1) * zeta(s-1)^2 * Product_{primes p} (1 - p^(3+m-3*s) + p^(2-2*s) + 2*p^(2+m-2*s) - p^(2+2*m-2*s) - 2*p^(1-s)).

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

%C Limit_{m->oo} c(m) = 6/Pi^2 = A059956.

%H Amiram Eldar, <a href="/A361268/b361268.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F Sum_{k=1..n} a(k) ~ c * zeta(3)^2 * n^4 / 4, where c = Product_{primes p} (1 - 1/p^2 - 2/p^3 + 2/p^4 + 1/p^6 - 1/p^7) = 0.47448576370894461600229128319633117903859559137234612880645471185501089953...

%F a(n) = A005361(n) * A361264(n).

%F Sum_{n>=1} 1/a(n) = Product_{p prime} (1 - log(1-1/p))/p^2 = 1.24331517732028787738... . - _Amiram Eldar_, Sep 01 2023

%t g[p_, e_] := 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)^2)[n], ", "))

%Y Cf. A005361, A059956, A203639, A361264, A361265.

%K nonn,easy,mult

%O 1,2

%A _Vaclav Kotesovec_, Mar 06 2023