A030231 Numbers with an even number of distinct prime factors.

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104, 106, 108, 111, 112, 115, 116, 117, 118, 119

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• easy
• nice

Comments

Gcd(A008472(a(n)), A007947(a(n)))=1; see A014963. - Labos Elemer, Mar 26 2003

Superset of A007774. - R. J. Mathar, Oct 23 2008

A076479(a(n)) = +1. - Reinhard Zumkeller, Jun 01 2013

Union of the rows of A125666 with even indices. - R. J. Mathar, Jul 19 2023

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• H. Helfgott and A. Ubis, Primos, paridad y análisis, arXiv:1812.08707 [math.NT], Dec. 2018.

Crossrefs

Cf. A028260, A030230, A123066.

Cf. A008472, A007947, A014963.

Cf. A007774, A076479.

Cf. A008683, A000005, A001221, A023900.

Programs

Haskell

a030231 n = a030231_list !! (n-1)
a030231_list = filter (even . a001221) [1..]
-- Reinhard Zumkeller, Mar 26 2013

Mathematica

Select[Range[200],EvenQ[PrimeNu[#]]&] (* Harvey P. Dale, Jun 22 2011 *)

PARI

j=[]; for(n=1,200,x=omega(n); if(Mod(x,2)==0,j=concat(j,n))); j

PARI

is(n)=omega(n)%2==0 \\ Charles R Greathouse IV, Sep 14 2015

Formula

From Benoit Cloitre, Dec 08 2002: (Start)

k such that Sum_{d|k} mu(d)*A000005(d) = (-1)^omega(k) = +1 where mu(d)=A008683(d), and omega(d)=A001221(d).

k such that A023900(k) > 0. (End)

Union of A007774, A033993, A074969,... - R. J. Mathar, Jul 22 2025

Extensions

Corrected by Dan Pritikin (pritikd(AT)muohio.edu), May 29 2002