A028260 - OEIS

%I #70 Apr 10 2025 12:28:11

%S 1,4,6,9,10,14,15,16,21,22,24,25,26,33,34,35,36,38,39,40,46,49,51,54,

%T 55,56,57,58,60,62,64,65,69,74,77,81,82,84,85,86,87,88,90,91,93,94,95,

%U 96,100,104,106,111,115,118,119,121,122,123,126,129,132,133,134

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

%C If k appears, p*k does not (p primes). - _Philippe Deléham_, Jun 10 2006

%C 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

%C A072978 is a subsequence. - _Reinhard Zumkeller_, Sep 20 2008

%C Quadratic residues of A191089(n) as n -> oo. - _Travis Scott_, Jan 14 2023

%H T. D. Noe, <a href="/A028260/b028260.txt">Table of n, a(n) for n = 1..10000</a>

%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/collectedpapers/Irregular/Irregular1.htm">Irregular numbers</a>, J. Indian Math. Soc., 5 (1913), 105-106; Coll. Papers 20-21.

%F A066829(a(n)) = 0. - _Reinhard Zumkeller_, Jun 26 2009

%F A001222(a(n)) mod 2 = 0. - _Reinhard Zumkeller_, Oct 05 2011

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

%p 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

%t Select[Range[200],EvenQ[PrimeOmega[#]]&] (* _Harvey P. Dale_, Aug 14 2011 *)

%t Select[Range@ 134, LiouvilleLambda@# > 0 &] (* _Robert G. Wilson v_, Jul 06 2012 *)

%o (Haskell)

%o a028260 n = a028260_list !! (n-1)

%o a028260_list = filter (even . a001222) [1..]

%o -- _Reinhard Zumkeller_, Oct 05 2011

%o (PARI) is(n)=bigomega(n)%2==0 \\ _Charles R Greathouse IV_, May 29 2013

%o (Python)

%o from math import isqrt, prod

%o from sympy import primerange, primepi, integer_nthroot

%o def A028260(n):

%o     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)))

%o     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)))

%o     m, k = n, f(n)

%o     while m != k: m, k = k, f(k)

%o     return m # _Chai Wah Wu_, Apr 10 2025

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

%K nonn,easy,nice

%O 1,2

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