OFFSET
1,8
COMMENTS
First differs from A365487 at n = 32.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(p^e) = 1 if e <= 2, a(p^e) = e+1 if e is odd and e > 1, and a(p^e) = e otherwise.
a(n) >= 1, with equality if and only if n is cubefree (A004709).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 3/p^3 - 1/p^5) = 1.69824776889117043774... .
MATHEMATICA
f[p_, e_] := If[e <= 2, 1, If[EvenQ[e], e, e + 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] <= 2, 1, if(!(f[i, 2]%2), f[i, 2], f[i, 2]+1)))};
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Dec 14 2023
STATUS
approved