A370786 - OEIS

%I #13 Mar 02 2024 03:51:05

%S 8,27,32,72,108,125,128,200,243,288,343,392,432,500,512,648,675,800,

%T 968,972,1125,1152,1323,1331,1352,1372,1568,1728,1800,2000,2048,2187,

%U 2197,2312,2592,2700,2888,3087,3125,3200,3267,3528,3872,3888,4232,4500,4563,4608

%N Powerful 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 * sqrt(x) + o(sqrt(x)) where c = (zeta(3/2)/zeta(3) - 1/zeta(3/2))/2 = 0.895230... .

%C The relative asymptotic density of this sequence within the powerful numbers is (1 - zeta(3)/(zeta(3/2)^2))/2 = 0.411930... .

%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 smaller than 1/2 when s is odd.

%H Amiram Eldar, <a href="/A370786/b370786.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, # > 1 &] && OddQ[Total[e]]]; Select[Range[2500], q]

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

%Y Intersection of A001694 and A026424.

%Y Complement of A370785 within A001694.

%Y A370788 is a subsequence.

%Y Cf. A002117, A078434, A090699.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Mar 02 2024