A382787 The product of exponents in the prime factorization of the numbers whose prime factorization contains exponents that are either 1 or even.

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Apr 05 2025

Keywords

• nonn
• easy

Comments

First differs from A368473 at n = 57.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005361, A013661, A065465, A335275, A368473.

Programs

Mathematica

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, # == 1 || EvenQ[#] &], Times @@ e, Nothing]]; Array[f, 150]

PARI

list(lim) = {my(e, ok); for(k = 1, lim, e = factor(k)[, 2]; ok = 1; for(i = 1, #e, if(e[i] > 1 && e[i]%2, ok = 0; break)); if(ok, print1(vecprod(e), ", ")));}

Formula

a(n) = A005361(A335275(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)^2 / A065465) * Product_{p prime} (1 - 1/p^2 - 2/p^3 + 3/p^4 - 1/p^6) = 1.568148713987289233406... .