A370788 Cubefull numbers with an odd number of prime factors (counted with multiplicity).

8, 27, 32, 125, 128, 243, 343, 432, 512, 648, 1331, 1728, 2000, 2048, 2187, 2197, 2592, 3125, 3888, 4913, 5000, 5488, 5832, 6859, 6912, 8000, 8192, 10125, 10368, 12167, 15552, 16807, 16875, 19208, 19683, 20000, 21296, 21952, 23328, 24389, 27000, 27648, 27783, 29791

Offset

1,1

Comments

Jakimczuk (2024) proved:

The number of terms that do not exceed x is N(x) = c * x^(1/3) / 2 + o(x^(1/3)) where c = A362974.

The relative asymptotic density of this sequence within the cubefull numbers is 1/2.

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 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, # > 2 &] && OddQ[Total[e]]]; Select[Range[30000], q]

Prog

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

Crossrefs

Intersection of A036966 and A026424.

Complement of A370787 within A036966.

Subsequence of A370786.

Cf. A362974.

Sequence in context: A376173 A335988 A097054 * A304291 A056824 A367804

Adjacent sequences: A370785 A370786 A370787 * A370789 A370790 A370791

Keyword

nonn,easy

Author

Amiram Eldar, Mar 02 2024

Status

approved