OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e <= 2, and 1 otherwise.
a(n) >= 1, with equality if and only if n is cubefull (A036966).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) - 1/p^(3*s-2) - 1/p^(3*s-1)). CHECK
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (p/(p+1) + 1/p - 1/p^4) = 0.63884633697952950095... .
MATHEMATICA
f[p_, e_] := If[e < 3, (p^(e+1)-1)/(p-1), 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] < 3, (f[i, 1]^(f[i, 2]+1)-1)/(f[i, 1]-1), 1)); }
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Oct 01 2023
STATUS
approved