OFFSET
1,4
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(p^2) = 2 and a(p^e) = e-1 if e != 2.
a(n) > 0 if and only if n is powerful (A001694).
Dirichlet g.f.: (zeta(2*s)*zeta(3*s)/zeta(6*s))^2.
Sum_{k=1..n} a(k) ~ (zeta(3/2)^2/(2*zeta(3)^2)) * sqrt(n) * (log(n) + 4*gamma - 2 + 6*zeta'(3/2)/zeta(3/2) - 12*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620).
MATHEMATICA
f[p_, e_] := If[e == 2, 2, e-1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecprod(apply(x -> if(x==2, 2, x-1), factor(n)[, 2]));
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Jan 19 2024
STATUS
approved