A370787 - OEIS

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A370787

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

1, 16, 64, 81, 216, 256, 625, 729, 864, 1000, 1024, 1296, 1944, 2401, 2744, 3375, 3456, 4000, 4096, 5184, 6561, 7776, 9261, 10000, 10648, 10976, 11664, 13824, 14641, 15625, 16000, 16384, 17496, 17576, 20736, 25000, 28561, 30375, 31104, 35937, 38416, 39304, 40000

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OFFSET

1,2

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 even 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 A028260.

Complement of A370788 within A036966.

Subsequence of A370785.

Cf. A362974.