cp's OEIS Frontend

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

%I A028260 #70 Apr 10 2025 12:28:11
%S A028260 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 A028260 55,56,57,58,60,62,64,65,69,74,77,81,82,84,85,86,87,88,90,91,93,94,95,
%U A028260 96,100,104,106,111,115,118,119,121,122,123,126,129,132,133,134
%N A028260 Numbers with an even number of prime divisors (counted with multiplicity); numbers k such that the Liouville function lambda(k) (A008836) is positive.
%C A028260 If k appears, p*k does not (p primes). - _Philippe Deléham_, Jun 10 2006
%C A028260 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 A028260 A072978 is a subsequence. - _Reinhard Zumkeller_, Sep 20 2008
%C A028260 Quadratic residues of A191089(n) as n -> oo. - _Travis Scott_, Jan 14 2023
%H A028260 T. D. Noe, <a href="/A028260/b028260.txt">Table of n, a(n) for n = 1..10000</a>
%H A028260 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 A028260 A066829(a(n)) = 0. - _Reinhard Zumkeller_, Jun 26 2009
%F A028260 A001222(a(n)) mod 2 = 0. - _Reinhard Zumkeller_, Oct 05 2011
%F A028260 Sum_{n>=1} 1/a(n)^s = (zeta(s)^2 + zeta(2*s))/(2*zeta(s)). - _Enrique Pérez Herrero_, Jul 06 2012
%p A028260 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 A028260 Select[Range[200],EvenQ[PrimeOmega[#]]&] (* _Harvey P. Dale_, Aug 14 2011 *)
%t A028260 Select[Range@ 134, LiouvilleLambda@# > 0 &] (* _Robert G. Wilson v_, Jul 06 2012 *)
%o A028260 (Haskell)
%o A028260 a028260 n = a028260_list !! (n-1)
%o A028260 a028260_list = filter (even . a001222) [1..]
%o A028260 -- _Reinhard Zumkeller_, Oct 05 2011
%o A028260 (PARI) is(n)=bigomega(n)%2==0 \\ _Charles R Greathouse IV_, May 29 2013
%o A028260 (Python)
%o A028260 from math import isqrt, prod
%o A028260 from sympy import primerange, primepi, integer_nthroot
%o A028260 def A028260(n):
%o A028260     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 A028260     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 A028260     m, k = n, f(n)
%o A028260     while m != k: m, k = k, f(k)
%o A028260     return m # _Chai Wah Wu_, Apr 10 2025
%Y A028260 Cf. A001222, A001358, A008836, A026424 (complement), A145784, A065043 (char. func).
%K A028260 nonn,easy,nice
%O A028260 1,2
%A A028260 Dan Asimov (dan(AT)research.att.com)