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

1,2

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.

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)

Mathematica

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

Prog

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

Crossrefs

Cf. A028260, A030230, A123066.

Cf. A008472, A007947, A014963.

Cf. A007774, A076479.

Cf. A008683, A000005, A001221, A023900.

Sequence in context: A324455 A327476 A007774 * A267114 A275665 A056760

Adjacent sequences: A030228 A030229 A030230 * A030232 A030233 A030234

Keyword

nonn,easy,nice

Author

David W. Wilson

Extensions

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

Status

approved