OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(p) = p^2 + 1 and a(p^e) = p^e + 1 for e >= 2.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(s-1) - 1/p^(2*s-3) + 1/p^(3*s-3) - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^6 - 2/p^7 + 1/p^8) = 0.73644353930922037459... .
MATHEMATICA
f[p_, e_] := If[e == 1, p^2 + 1, p^e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1 + f[i, 1]^2, 1 + f[i, 1]^f[i, 2])); }
CROSSREFS
KEYWORD
AUTHOR
Amiram Eldar, Jan 30 2024
STATUS
approved