OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(p^e) = e+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/p^(2*s) - 2/p^(3*s)).
From Vaclav Kotesovec, Oct 01 2023: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 3/p^(3*s) + 2/p^(4*s)).
Let f(s) = Product_{p prime} (1 - 3/p^(3*s) + 2/p^(4*s)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 3/p^3 + 2/p^4) = 0.66219033176371496870504912254207846719824904470940603905284774924086...,
f'(1) = f(1) * Sum_{p prime} (9*p - 8) * log(p) / (p^4 - 3*p + 2) = f(1) * 1.04316863044761953555286128194165251303791613504188623828521117799260...
and gamma is the Euler-Mascheroni constant A001620. (End)
MATHEMATICA
f[p_, e_] := If[e < 3, e+1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecprod(apply(x -> if(x < 3, x+1, 1), factor(n)[, 2]));
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Oct 01 2023
STATUS
approved