OFFSET
1,2
COMMENTS
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(p) = p + 1, and a(p^e) = (p^(e+1) - 1)/(p - 1) - p^2 for e >= 2.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(3*s-3)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^3-p)) = 1.231291... (A065487).
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p^2; f[p_, 1] := p + 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, f[i, 1] + 1, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - f[i, 1]^2)); }
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Oct 21 2023
STATUS
approved