A370786 - OEIS

A370786

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

8, 27, 32, 72, 108, 125, 128, 200, 243, 288, 343, 392, 432, 500, 512, 648, 675, 800, 968, 972, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1728, 1800, 2000, 2048, 2187, 2197, 2312, 2592, 2700, 2888, 3087, 3125, 3200, 3267, 3528, 3872, 3888, 4232, 4500, 4563, 4608

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OFFSET

1,1

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 = 0.895230... .

The relative asymptotic density of this sequence within the powerful numbers is (1 - zeta(3)/(zeta(3/2)^2))/2 = 0.411930... .

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

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]]}, AllTrue[e, # > 1 &] && OddQ[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 A026424.

Complement of A370785 within A001694.

A370788 is a subsequence.

Cf. A002117, A078434, A090699.