cp's OEIS Frontend

A030231 Numbers with an even number of distinct prime factors.

%I A030231 #50 Jul 22 2025 14:21:38
%S A030231 1,6,10,12,14,15,18,20,21,22,24,26,28,33,34,35,36,38,39,40,44,45,46,
%T A030231 48,50,51,52,54,55,56,57,58,62,63,65,68,69,72,74,75,76,77,80,82,85,86,
%U A030231 87,88,91,92,93,94,95,96,98,99,100,104,106,108,111,112,115,116,117,118,119
%N A030231 Numbers with an even number of distinct prime factors.
%C A030231 Gcd(A008472(a(n)), A007947(a(n)))=1; see A014963. - _Labos Elemer_, Mar 26 2003
%C A030231 Superset of A007774. - _R. J. Mathar_, Oct 23 2008
%C A030231 A076479(a(n)) = +1. - _Reinhard Zumkeller_, Jun 01 2013
%C A030231 Union of the rows of A125666 with even indices. - _R. J. Mathar_, Jul 19 2023
%H A030231 T. D. Noe, <a href="/A030231/b030231.txt">Table of n, a(n) for n = 1..1000</a>
%H A030231 H. Helfgott and A. Ubis, <a href="https://arxiv.org/abs/1812.08707">Primos, paridad y análisis</a>, arXiv:1812.08707 [math.NT], Dec. 2018.
%F A030231 From _Benoit Cloitre_, Dec 08 2002: (Start)
%F A030231 k such that Sum_{d|k} mu(d)*A000005(d) = (-1)^omega(k) = +1 where mu(d)=A008683(d), and omega(d)=A001221(d).
%F A030231 k such that A023900(k) > 0. (End)
%F A030231 Union of A007774, A033993, A074969,... - _R. J. Mathar_, Jul 22 2025
%t A030231 Select[Range[200],EvenQ[PrimeNu[#]]&] (* _Harvey P. Dale_, Jun 22 2011 *)
%o A030231 (PARI) j=[]; for(n=1,200,x=omega(n); if(Mod(x,2)==0,j=concat(j,n))); j
%o A030231 (PARI) is(n)=omega(n)%2==0 \\ _Charles R Greathouse IV_, Sep 14 2015
%o A030231 (Haskell)
%o A030231 a030231 n = a030231_list !! (n-1)
%o A030231 a030231_list = filter (even . a001221) [1..]
%o A030231 -- _Reinhard Zumkeller_, Mar 26 2013
%Y A030231 Cf. A028260, A030230, A123066.
%Y A030231 Cf. A008472, A007947, A014963.
%Y A030231 Cf. A007774, A076479.
%Y A030231 Cf. A008683, A000005, A001221, A023900.
%K A030231 nonn,easy,nice
%O A030231 1,2
%A A030231 _David W. Wilson_
%E A030231 Corrected by Dan Pritikin (pritikd(AT)muohio.edu), May 29 2002