A360909 Multiplicative with a(p^e) = 3*e + 2.

1, 5, 5, 8, 5, 25, 5, 11, 8, 25, 5, 40, 5, 25, 25, 14, 5, 40, 5, 40, 25, 25, 5, 55, 8, 25, 11, 40, 5, 125, 5, 17, 25, 25, 25, 64, 5, 25, 25, 55, 5, 125, 5, 40, 40, 25, 5, 70, 8, 40, 25, 40, 5, 55, 25, 55, 25, 25, 5, 200, 5, 25, 40, 20, 25, 125, 5, 40, 25, 125

Offset: 1

Author

Vaclav Kotesovec, Feb 25 2023

Keywords

• nonn
• mult

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), A360997 (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), this sequence (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).

Programs

Mathematica

a[n_] := Times @@ ((3*Last[#] + 2) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)

PARI

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

Formula

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

Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 7/p^(2*s) + 11/p^(3*s) - 6/p^(4*s) + 1/p^(5*s)), (with a product that converges for s=1).

Sum_{k=1..n} a(k) ~ c * n * log(n)^4 / 24, where c = Product_{primes p} (1 - 7/p^2 + 11/p^3 - 6/p^4 + 1/p^5) = 0.091414252314317101861531055690354339957600046..., more precise (but very complicated) asymptotics can be obtained (in Mathematica notation) as Residue[Zeta[s]^5 * f[s] * n^s / s, {s, 1}], where f[s] = Product_{primes p} (1 - 7/p^(2*s) + 11/p^(3*s) - 6/p^(4*s) + 1/p^(5*s)).