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A370787
Cubefull numbers with an even number of prime factors (counted with multiplicity).
3
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
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
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Mar 02 2024
STATUS
approved