|
|
A360909
|
|
Multiplicative with a(p^e) = 3*e + 2.
|
|
5
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
LINKS
|
|
|
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)).
|
|
MATHEMATICA
|
a[n_] := Times @@ ((3*Last[#] + 2) & /@ FactorInteger[n]); a[1] = 1; Array[a, 100] (* Amiram Eldar, Feb 25 2023 *)
|
|
PROG
|
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1+3*X-X^2)/(1-X)^2)[n], ", "))
|
|
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).
|
|
KEYWORD
|
nonn,mult
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|