A360910 Multiplicative with a(p^e) = 3*e - 1.
1, 2, 2, 5, 2, 4, 2, 8, 5, 4, 2, 10, 2, 4, 4, 11, 2, 10, 2, 10, 4, 4, 2, 16, 5, 4, 8, 10, 2, 8, 2, 14, 4, 4, 4, 25, 2, 4, 4, 16, 2, 8, 2, 10, 10, 4, 2, 22, 5, 10, 4, 10, 2, 16, 4, 16, 4, 4, 2, 20, 2, 4, 10, 17, 4, 8, 2, 10, 4, 8, 2, 40, 2, 4, 10, 10, 4, 8, 2
Offset: 1
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Programs
-
Mathematica
a[n_] := Times @@ ((3*Last[#] - 1) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)
-
PARI
for(n=1, 100, print1(direuler(p=2, n, (1+2*X^2)/(1-X)^2)[n], ", "))
Formula
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 2/p^(2*s)).
Let f(s) = Product_{primes p} (1 + 2/p^(2*s)), then Sum_{k=1..n} a(k) ~ n*(f(1)*(log(n) + 2*gamma - 1) + f'(1)), where f(1) = Product_{primes p} (1 + 2/p^2) = 2.1908700855532557963501937947188223715671192999357721091330157224657649571..., f'(1) = f(1) * Sum_{primes p} (-4*log(p)/(p^2 + 2)) = -3.559220569509264750413960031425742000438433285978558703470289340806139902... and gamma is the Euler-Mascheroni constant A001620.