OFFSET

1,2

COMMENTS

Jakimczuk (2024) proved:

The number of terms that do not exceed x is N(x) = c * sqrt(x) + o(sqrt(x)) where c = (zeta(3/2)/zeta(3) + 1/zeta(3/2))/2 = 1.278023... .

The relative asymptotic density of this sequence within the powerful numbers is (1 + zeta(3)/(zeta(3/2)^2))/2 = 0.588069... .

In general, the relative asymptotic density of the s-full numbers (numbers whose exponents in their prime factorization are all >= s) with an even number of prime factors (counted with multiplicity) within the s-full numbers is larger than 1/2 when s is even.

LINKS

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

Rafael Jakimczuk, Arithmetical Functions over the Powerful Part of an Integer, ResearchGate, 2024.

MATHEMATICA

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, n == 1 || AllTrue[e, # > 1 &] && EvenQ[Total[e]]]; Select[Range[2500], q]

PROG

(PARI) is(n) = {my(e = factor(n)[, 2]); n == 1 || (vecmin(e) > 1 && !(vecsum(e)%2)); }

CROSSREFS

KEYWORD

nonn,easy

AUTHOR

Amiram Eldar, Mar 02 2024

STATUS

approved