A370785 Powerful numbers with an even number of prime factors (counted with multiplicity).

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 216, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 864, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 1944, 2025, 2116, 2209

Offset: 1

Author

Amiram Eldar, Mar 02 2024

Keywords

• nonn
• easy

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.

Crossrefs

Intersection of A001694 and A028260.

Complement of A370786 within A001694.

A370787 is a subsequence.

Cf. A002117, A078434, A090699.

Programs

Mathematica

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

PARI

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