%I M4065 N1685
%N Numbers n where total number of 1-bits in the exponents of their prime factorization is even; a 2-way 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 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/CMB-1959-013-x">On some two way classifications of integers</a>, Canad. Math. Bull. 2 (1959), 85-89.
%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, EvenQ[ Count[ Flatten[ IntegerDigits[#, 2]& /@ Transpose[ FactorInteger[#]][]], 1]]&] // Prepend[#, 1]& (* _Jean-François Alcover_, Apr 11 2013, after _Harvey P. Dale_ *)
%o a000379 n = a000379_list !! (n-1)
%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 (MATCHING-POS 1 1 (COMPOSE even? A064547)))
%o (define A000379 (MATCHING-POS 1 1 (lambda (n) (even? (A000120 (A268387 n))))))
%o ;; Both require also my IntSeq-library. - _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.
%A _N. J. A. Sloane_
%E Edited by _N. J. A. Sloane_, Dec 20 2007, to restore the original definition.