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Cubefull numbers with an even number of prime factors (counted with multiplicity).
3

%I #12 Mar 02 2024 03:44:36

%S 1,16,64,81,216,256,625,729,864,1000,1024,1296,1944,2401,2744,3375,

%T 3456,4000,4096,5184,6561,7776,9261,10000,10648,10976,11664,13824,

%U 14641,15625,16000,16384,17496,17576,20736,25000,28561,30375,31104,35937,38416,39304,40000

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

%C Jakimczuk (2024) proved:

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

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

%C 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.

%H Amiram Eldar, <a href="/A370787/b370787.txt">Table of n, a(n) for n = 1..10000</a>

%H Rafael Jakimczuk, <a href="http://dx.doi.org/10.13140/RG.2.2.12174.13124">Arithmetical Functions over the Powerful Part of an Integer</a>, ResearchGate, 2024.

%t q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, # > 2 &] && OddQ[Total[e]]]; Select[Range[30000], q]

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

%Y Intersection of A036966 and A028260.

%Y Complement of A370788 within A036966.

%Y Subsequence of A370785.

%Y Cf. A362974.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Mar 02 2024