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A000379 Numbers where total number of 1-bits in the exponents of their prime factorization is even; a 2-way classification of integers: complement of A000028.

%I A000379 M4065 N1685 #75 Feb 16 2025 08:32:21
%S A000379 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 A000379 48,50,51,52,55,57,58,62,63,64,65,68,69,74,75,76,77,80,82,85,86,87,91,
%U A000379 92,93,94,95,98,99,100,106,111,112,115,116,117,118,119,120,122,123,124,125,129
%N A000379 Numbers 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 A000379 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 A000379 See A000028 for precise definition, Maple program, etc.
%C A000379 The sequence contains products of even number of distinct terms of A050376. - _Vladimir Shevelev_, May 04 2010
%C A000379 From _Vladimir Shevelev_, Oct 28 2013: (Start)
%C A000379 Numbers m such that the infinitary Möbius function (A064179) of m equals 1. (This follows from the definition of A064179.)
%C A000379 A number m is in the sequence iff the number k = k(m) of terms of A050376 that divide m with odd maximal exponent is even (see example).
%C A000379 (End)
%C A000379 Numbers k for which A064547(k) [or equally, A268386(k)] is even. Numbers k for which A010060(A268387(k)) = 0. - _Antti Karttunen_, Feb 09 2016
%C A000379 The sequence is closed under the commutative binary operation A059897(.,.). As integers are self-inverse under A059897(.,.), it therefore forms a subgroup of the positive integers considered as a group under A059897(.,.). Specifically (expanding on the comment above dated May 04 2010) it is the subgroup of even length words in A050376, which is the group's lexicographically earliest ordered minimal set of generators. A000028, the set of odd length words in A050376, is its complementary coset. - _Peter Munn_, Nov 01 2019
%C A000379 From _Amiram Eldar_, Oct 02 2024: (Start)
%C A000379 Numbers whose number of infinitary divisors (A037445) is a square.
%C A000379 Numbers whose exponentially odious part (A367514) has an even number of distinct prime factors, i.e., numbers k such that A092248(A367514(k)) = 0. (End)
%D A000379 Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 22.
%D A000379 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000379 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000379 N. J. A. Sloane, <a href="/A000379/b000379.txt">Table of n, a(n) for n = 1..10000</a>
%H A000379 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.
%H A000379 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Group.html">Group</a>.
%H A000379 Wikipedia, <a href="https://en.wikipedia.org/wiki/Generating_set_of_a_group">Generating set of a group</a>.
%e A000379 If m = 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 A000379 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 A000379 (Haskell)
%o A000379 a000379 n = a000379_list !! (n-1)
%o A000379 a000379_list = filter (even . sum . map a000120 . a124010_row) [1..]
%o A000379 -- _Reinhard Zumkeller_, Oct 05 2011
%o A000379 (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 A000379 (Scheme, two variants)
%o A000379 (define A000379 (MATCHING-POS 1 1 (COMPOSE even? A064547)))
%o A000379 (define A000379 (MATCHING-POS 1 1 (lambda (n) (even? (A000120 (A268387 n))))))
%o A000379 ;; Both require also my IntSeq-library. - _Antti Karttunen_, Feb 09 2016
%Y A000379 Subsequences: A030229, A238748, A262675, A268390.
%Y A000379 Subsequence of A268388 (apart from the initial 1).
%Y A000379 Complement: A000028.
%Y A000379 Sequences used in definitions of this sequence: A133008, A050376, A059897, A064179, A064547, A124010 (prime exponents), A268386, A268387, A010060.
%Y A000379 Other 2-way classifications: A000069/A001969 (to which A000120 and A010060 are relevant), A000201/A001950.
%Y A000379 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.
%Y A000379 Cf. A030231, A037445, A092248, A367514.
%K A000379 nonn,easy,nice
%O A000379 1,2
%A A000379 _N. J. A. Sloane_
%E A000379 Edited by _N. J. A. Sloane_, Dec 20 2007, to restore the original definition.