A370785 - OEIS

%I #11 Mar 02 2024 03:48:08

%S 1,4,9,16,25,36,49,64,81,100,121,144,169,196,216,225,256,289,324,361,

%T 400,441,484,529,576,625,676,729,784,841,864,900,961,1000,1024,1089,

%U 1156,1225,1296,1369,1444,1521,1600,1681,1764,1849,1936,1944,2025,2116,2209

%N Powerful 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 * sqrt(x) + o(sqrt(x)) where c = (zeta(3/2)/zeta(3) + 1/zeta(3/2))/2 = 1.278023... .

%C The relative asymptotic density of this sequence within the powerful numbers is (1 + zeta(3)/(zeta(3/2)^2))/2 = 0.588069... .

%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 larger than 1/2 when s is even.

%H Amiram Eldar, <a href="/A370785/b370785.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]]}, n == 1 || AllTrue[e, # > 1 &] && EvenQ[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 A028260.

%Y Complement of A370786 within A001694.

%Y A370787 is a subsequence.

%Y Cf. A002117, A078434, A090699.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Mar 02 2024