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

1,2

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

Intersection of A001694 and A028260.

Complement of A370786 within A001694.

A370787 is a subsequence.

Cf. A002117, A078434, A090699.

Sequence in context: A194148 A162497 A216500 * A069821 A331221 A174452

Adjacent sequences: A370782 A370783 A370784 * A370786 A370787 A370788

Keyword

nonn,easy

Author

Amiram Eldar, Mar 02 2024

Status

approved