%I M4065 N1685
%S 1,6,8,10,12,14,15,18,20,21,22,26,27,28,32,33,34,35,36,38,39,44,45,46,
%T 48,50,51,52,55,57,58,62,63,64,65,68,69,74,75,76,77,80,82,85,86,87,91,
%U 92,93,94,95,98,99,100,106,111,112,115,116,117,118,119,120,122,123,124,125,129
%N Numbers n where total number of 1bits in the exponents of their prime factorization is even; a 2way classification of integers: complement of A000028.
%C This sequence and A000028 (its complement) give the unique solution to the problem of splitting the positive integers into two classes in such a way that products of pairs of distinct elements from either class occur with the same multiplicities [Lambek and Moser]. Cf. A000069, A001969.
%C See A000028 for precise definition, Maple program, etc.
%C The sequence contains products of even number of distinct terms of A050376.  _Vladimir Shevelev_, May 04 2010
%C From _Vladimir Shevelev_, Oct 28 2013: (Start)
%C Or infinitary Möbius function (A064179) of n equals 1. (This follows from the definition of A064179.)
%C A number n is in the sequence iff the number k=k(n) of terms of A050376 that divide n with odd maximal exponent is even (see example).
%C (End)
%C Numbers n for which A064547(n) [or equally, A268386(n)] is even. Numbers n for which A010060(A268387(n)) = 0.  _Antti Karttunen_, Feb 09 2016
%D J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 22.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H N. J. A. Sloane, <a href="/A000379/b000379.txt">Table of n, a(n) for n = 1..10000</a>
%H J. Lambek and L. Moser, <a href="http://dx.doi.org/10.4153/CMB1959013x">On some two way classifications of integers</a>, Canad. Math. Bull. 2 (1959), 8589.
%e If n=120, then the maximal exponent of 2 that divides 120 is 3, for 3 it is 1, for 4 it is 1, for 5 it is 1. Thus k(120)=4 and 120 is a term.  _Vladimir Shevelev_, Oct 28 2013
%t Select[ Range[130], EvenQ[ Count[ Flatten[ IntegerDigits[#, 2]& /@ Transpose[ FactorInteger[#]][[2]]], 1]]&] // Prepend[#, 1]& (* _JeanFrançois Alcover_, Apr 11 2013, after _Harvey P. Dale_ *)
%o (Haskell)
%o a000379 n = a000379_list !! (n1)
%o a000379_list = filter (even . sum . map a000120 . a124010_row) [1..]
%o  _Reinhard Zumkeller_, Oct 05 2011
%o (PARI) is(n)=my(f=factor(n)[,2]); sum(i=1,#f,hammingweight(f[i]))%2==0 \\ _Charles R Greathouse IV_, Aug 31 2013
%o (Scheme, two variants)
%o (define A000379 (MATCHINGPOS 1 1 (COMPOSE even? A064547)))
%o (define A000379 (MATCHINGPOS 1 1 (lambda (n) (even? (A000120 (A268387 n))))))
%o ;; Both require also my IntSeqlibrary.  _Antti Karttunen_, Feb 09 2016
%Y Subsequences: A238748, A268390.
%Y Subsequence of A268388 (apart from the initial 1).
%Y Cf. A133008, A000028 (complement), A000120, A000201, A001950, A010060, A064547, A268386, A268387, A124010 (prime exponents).
%Y This is different from A123240 (e.g., does not contain 180). The first difference occurs already at n=31, where A123240(31) = 60, a value which does not occur here, as a(31+1) = 62. The same is true with respect to A131181, as A131181(31) = 60.
%K nonn,easy,nice
%O 1,2
%A _N. J. A. Sloane_
%E Edited by _N. J. A. Sloane_, Dec 20 2007, to restore the original definition.
