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

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.

Sequence in context: A365263 A062320 A233330 * A322449 A117453 A374291

Adjacent sequences: A370784 A370785 A370786 * A370788 A370789 A370790

Keyword

nonn,easy

Author

Amiram Eldar, Mar 02 2024

Status

approved