A028260 Numbers with an even number of prime divisors (counted with multiplicity); numbers k such that the Liouville function lambda(k) (A008836) is positive.

1, 4, 6, 9, 10, 14, 15, 16, 21, 22, 24, 25, 26, 33, 34, 35, 36, 38, 39, 40, 46, 49, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 69, 74, 77, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 100, 104, 106, 111, 115, 118, 119, 121, 122, 123, 126, 129, 132, 133, 134

Offset: 1

Author

Dan Asimov (dan(AT)research.att.com)

Keywords

• nonn
• easy
• nice

Comments

If k appears, p*k does not (p primes). - Philippe Deléham, Jun 10 2006

The product of any two terms of this sequence, or any two terms of the complement of this sequence (A026424), is a term of this sequence. The product of a term of this sequence and a term of A026424 is a term of A026424. The primitive terms of this sequence are the semiprimes (A001358). - Franklin T. Adams-Watters, Nov 27 2006

A072978 is a subsequence. - Reinhard Zumkeller, Sep 20 2008

Quadratic residues of A191089(n) as n -> oo. - Travis Scott, Jan 14 2023

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• S. Ramanujan, Irregular numbers, J. Indian Math. Soc., 5 (1913), 105-106; Coll. Papers 20-21.

Crossrefs

Cf. A001222, A001358, A008836, A026424 (complement), A145784, A065043 (char. func).

Programs

Haskell

a028260 n = a028260_list !! (n-1)
a028260_list = filter (even . a001222) [1..]
-- Reinhard Zumkeller, Oct 05 2011

Maple

with(numtheory); A028260 := proc(n) option remember: local k: if(n=1)then return 1: fi: for k from procname(n-1)+1 do if(bigomega(k) mod 2=0)then return k: fi: od: end: seq(A028260(n),n=1..63); # Nathaniel Johnston, May 27 2011

Mathematica

Select[Range[200],EvenQ[PrimeOmega[#]]&] (* Harvey P. Dale, Aug 14 2011 *)
Select[Range@ 134, LiouvilleLambda@# > 0 &] (* Robert G. Wilson v, Jul 06 2012 *)

PARI

is(n)=bigomega(n)%2==0 \\ Charles R Greathouse IV, May 29 2013

Python

from math import isqrt, prod
from sympy import primerange, primepi, integer_nthroot
def A028260(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-1-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,m)) for m in range(2,x.bit_length()+1,2)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Apr 10 2025

Formula

A066829(a(n)) = 0. - Reinhard Zumkeller, Jun 26 2009

A001222(a(n)) mod 2 = 0. - Reinhard Zumkeller, Oct 05 2011

Sum_{n>=1} 1/a(n)^s = (zeta(s)^2 + zeta(2*s))/(2*zeta(s)). - Enrique Pérez Herrero, Jul 06 2012