A370788 - OEIS

%I #11 Mar 02 2024 03:50:59

%S 8,27,32,125,128,243,343,432,512,648,1331,1728,2000,2048,2187,2197,

%T 2592,3125,3888,4913,5000,5488,5832,6859,6912,8000,8192,10125,10368,

%U 12167,15552,16807,16875,19208,19683,20000,21296,21952,23328,24389,27000,27648,27783,29791

%N Cubefull numbers with an odd 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 odd number of prime factors (counted with multiplicity) within the s-full numbers is 1/2 when s is odd.

%H Amiram Eldar, <a href="/A370788/b370788.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 A026424.

%Y Complement of A370787 within A036966.

%Y Subsequence of A370786.

%Y Cf. A362974.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Mar 02 2024