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A000379 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.
(Formerly M4065 N1685)

%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 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 (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/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[130], EvenQ[ Count[ Flatten[ IntegerDigits[#, 2]& /@ Transpose[ FactorInteger[#]][[2]]], 1]]&] // Prepend[#, 1]& (* _Jean-François Alcover_, Apr 11 2013, after _Harvey P. Dale_ *)

%o (Haskell)

%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.

%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.

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Last modified August 20 03:33 EDT 2019. Contains 326139 sequences. (Running on oeis4.)