OFFSET
1,4
COMMENTS
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
a(n) >= 1 with equality if and only if n is squarefree (A005117).
Multiplicative with a(p) = 1 and a(p^e) = p^(2*floor(e/2)) + 1 for e >= 2.
Dirichlet g.f.: zeta(s) * zeta(2*s-2) / zeta(4*s-2).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/(3*zeta(4)) = 30*zeta(3/2)/Pi^4 = 0.804557969165... .
MATHEMATICA
f[p_, e_] := If[e == 1, 1, p^(2*Floor[e/2]) + 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, 1 + f[i, 1]^(2*(f[i, 2]\2)))); }
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Sep 03 2023
STATUS
approved