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A026424 Number of prime divisors (counted with multiplicity) is odd; Liouville function lambda(n) (A008836) is negative. 50

%I

%S 2,3,5,7,8,11,12,13,17,18,19,20,23,27,28,29,30,31,32,37,41,42,43,44,

%T 45,47,48,50,52,53,59,61,63,66,67,68,70,71,72,73,75,76,78,79,80,83,89,

%U 92,97,98,99,101,102,103,105,107,108,109,110,112

%N Number of prime divisors (counted with multiplicity) is odd; Liouville function lambda(n) (A008836) is negative.

%C Neither this sequence nor its complement (A028260) contains any infinite arithmetic progression. - _Franklin T. Adams-Watters_, Sep 05 2008

%C A066829(a(n)) = 1. - _Reinhard Zumkeller_, Jun 26 2009

%C These numbers can be generated by the sieving process described in A066829. - _Reinhard Zumkeller_, Jul 01 2009

%C Lexicographically earliest sequence of distinct nonnegative integers with no term being the product of any two not necessarily distinct terms. The equivalent sequence for addition/subtraction is A005408 (the odd numbers), for exponentiation is A259444, and for binary exclusive OR is A000069. - _Peter Munn_, Mar 16 2018

%C The equivalent lexicographically earliest sequence with no term being the product of any two distinct terms is A026416. A000028 is similarly the equivalent sequence when A059897 is used as multiplicative operator in place of standard integer multiplication. - _Peter Munn_, Mar 16 2019

%H T. D. Noe, <a href="/A026424/b026424.txt">Table of n, a(n) for n = 1..10000</a>

%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper4/page1.htm">Irregular numbers</a>, J. Indian Math. Soc., 5 (1913), 105-106; Coll. Papers 20-21.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>

%H <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a> - _Reinhard Zumkeller_, Jul 01 2009

%F Sum 1/a(n)^m = (zeta(m)^2-zeta(2m))/(2*zeta(m)), Dirichlet g.f. of A066829. - Ramanujan.

%F n>=2 is in sequence if n is not the product of two smaller elements. - _David W. Wilson_, May 06 2005

%F A001222(a(n)) mod 2 = 1. - _Reinhard Zumkeller_, Oct 05 2011

%F Union of A000040, A014612, A014614, A046308 etc. - _R. J. Mathar_, Jul 09 2012

%p isA026424 := proc(n)

%p if type(numtheory[bigomega](n) ,'odd') then

%p true;

%p else

%p false;

%p end if;

%p end proc:

%p A026424 := proc(n)

%p option remember;

%p if n =1 then

%p 2;

%p else

%p for a from procname(n-1)+1 do

%p if isA026424(a) then

%p return a;

%p end if;

%p end do:

%p end if;

%p end proc: # _R. J. Mathar_, May 25 2017

%t Select[Range[2, 112], OddQ[Total[FactorInteger[#]][[2]]] &] (* _T. D. Noe_, May 07 2011 *)

%t (* From version 7 on *) Select[Range[2, 112], LiouvilleLambda[#] == -1 &] (* _Jean-Fran├žois Alcover_, Aug 19 2013 *)

%o (Haskell)

%o a026424 n = a026424_list !! (n-1)

%o a026424_list = filter (odd . a001222) [1..]

%o -- _Reinhard Zumkeller_, Oct 05 2011

%o (PARI) is(n)=bigomega(n)%2 \\ _Charles R Greathouse IV_, Sep 16 2015

%Y Cf. A008836, A028260 (complement).

%Y Apart from initial term, same as A026422.

%Y Cf. A026416 and cross-references therein.

%Y Cf. A000028, A000069, A005408, A259444.

%K nonn,easy,nice

%O 1,1

%A _N. J. A. Sloane_.

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Last modified November 19 22:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)