A370080 The product of the even exponents of the prime factorization of n.
1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
-
Mathematica
f[p_, e_] := If[EvenQ[e], e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
-
PARI
a(n) = vecprod(apply(x -> if(x%2, 1, x), factor(n)[, 2]));
Formula
Multiplicative with a(p^e) = e if e is even, and 1 if e is odd.
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s - 1/p^(3*s) + 1/p^(4*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)^2 * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 2/p^4 - 1/p^5) = 1.53318063378623623841... .
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + (p^(2*s) + 1)/(p^s*(p^s - 1)*(p^s + 1)^2)). - Vaclav Kotesovec, Feb 11 2024