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%I A000028 M0520 N0187 #80 Feb 14 2025 19:26:18
%S A000028 2,3,4,5,7,9,11,13,16,17,19,23,24,25,29,30,31,37,40,41,42,43,47,49,53,
%T A000028 54,56,59,60,61,66,67,70,71,72,73,78,79,81,83,84,88,89,90,96,97,101,
%U A000028 102,103,104,105,107,108,109,110,113,114,121,126,127,128,130,131,132,135,136,137
%N A000028 Let k = p_1^e_1 p_2^e_2 p_3^e_3 ... be the prime factorization of n. Sequence gives k such that the sum of the numbers of 1's in the binary expansions of e_1, e_2, e_3, ... is odd.
%C A000028 This sequence and A000379 (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 A000028 Contains (for example) 180, so is different from A123193. - _Max Alekseyev_, Sep 20 2007
%C A000028 The sequence contains products of odd number of distinct terms of A050376. - _Vladimir Shevelev_, May 04 2010
%C A000028 From _Vladimir Shevelev_, Oct 28 2013: (Start)
%C A000028 Numbers m such that infinitary Moebius function of m (A064179) equals -1. This follows from the definition of A064179.
%C A000028 Number m is in the sequence if and only if the number k = k(m) of terms of A050376 which divide m with odd maximal exponent is odd.
%C A000028 For example, if m = 96, then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1, for 4 it is 2, for 16 it is 1. Thus k(96) = 3 and 96 is a term.
%C A000028 (End)
%C A000028 Positions of odd terms in A064547, A268386 and A293439. - _Antti Karttunen_, Nov 09 2017
%C A000028 Lexicographically earliest sequence of distinct nonnegative integers such that no term is the A059897 product of 2 terms. (A059897 can be considered as a multiplicative operator related to the Fermi-Dirac factorization of numbers described in A050376.) Specifying that the A059897 product be of 2 distinct terms leaves the sequence unchanged. The equivalent sequences using standard integer multiplication are A026416 (with the 2 terms specified as distinct) and A026424 (otherwise). - _Peter Munn_, Mar 16 2019
%C A000028 From _Amiram Eldar_, Oct 02 2024: (Start)
%C A000028 Numbers whose number of infinitary divisors (A037445) is not a square.
%C A000028 Numbers whose exponentially odious part (A367514) has an odd number of distinct prime factors, i.e., numbers k such that A092248(A367514(k)) = 1. (End)
%D A000028 Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 22.
%D A000028 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000028 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000028 N. J. A. Sloane, <a href="/A000028/b000028.txt">Table of n, a(n) for n = 1..10000</a>
%H A000028 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 A000028 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%e A000028 If k = 96 then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1. 5 in binary is 101_2 and has so has a sum of binary digits of 1 + 0 + 1 = 2. 1 in binary is 1_2 and so has a sum of binary digits of 1. Thus the sum of digits of binary exponents is 2 + 1 = 3 which is odd and so 96 is a term. - _Vladimir Shevelev_, Oct 28 2013, edited by _David A. Corneth_, Mar 20 2019
%p A000028 (Maple program from _N. J. A. Sloane_, Dec 20 2007) expts:=proc(n) local t1,t2,t3,t4,i; if n=1 then RETURN([0]); fi; if isprime(n) then RETURN([1]); fi; t1:=ifactor(n); if nops(factorset(n))=1 then RETURN([op(2,t1)]); fi; t2:=nops(t1); t3:=[]; for i from 1 to t2 do t4:=op(i,t1); if nops(t4) = 1 then t3:=[op(t3),1]; else t3:=[op(t3),op(2,t4)]; fi; od; RETURN(t3); end; # returns a list of the exponents e_1, e_2, ...
%p A000028 A000120 := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: # returns weight of binary expansion
%p A000028 LamMos:= proc(n) local t1,t2,t3,i; t1:=expts(n); add( A000120(t1[i]),i=1..nops(t1)); end; # returns sum of weights of exponents
%p A000028 M:=400; t0:=[]; t1:=[]; for n from 1 to M do if LamMos(n) mod 2 = 0 then t0:=[op(t0),n] else t1:=[op(t1),n]; fi; od: t0; t1; # t0 is A000379, t1 is the present sequence
%t A000028 iMoebiusMu[ n_ ] := Switch[ MoebiusMu[ n ], 1, 1, -1, -1, 0, If[ OddQ[ Plus@@ (DigitCount[ Last[ Transpose[ FactorInteger[ n ] ] ], 2, 1 ]) ], -1, 1 ] ]; q=Select[ Range[ 20000 ],iMoebiusMu[ # ]===-1& ] (* _Wouter Meeussen_, Dec 21 2007 *)
%t A000028 Rest[Select[Range[150],OddQ[Count[Flatten[IntegerDigits[#,2]&/@ Transpose[ FactorInteger[#]][[2]]],1]]&]] (* _Harvey P. Dale_, Feb 25 2012 *)
%o A000028 (Haskell)
%o A000028 a000028 n = a000028_list !! (n-1)
%o A000028 a000028_list = filter (odd . sum . map a000120 . a124010_row) [1..]
%o A000028 -- _Reinhard Zumkeller_, Oct 05 2011
%o A000028 (PARI) is(n)=my(f=factor(n)[,2]); sum(i=1,#f,hammingweight(f[i]))%2 \\ _Charles R Greathouse IV_, Aug 31 2013
%Y A000028 Cf. A133008, A000379 (complement), A000120 (binary weight function), A064547; also A066724, A026477, A050376, A084400, A268386, A293439.
%Y A000028 Note that A000069 and A001969, also A000201 and A001950 give other decompositions of the integers into two classes.
%Y A000028 Cf. A124010 (prime exponents).
%Y A000028 Cf. A026416, A026424, A059897.
%Y A000028 Cf. A030230, A037445, A092248, A367514.
%K A000028 nonn,nice,easy
%O A000028 1,1
%A A000028 _N. J. A. Sloane_, _Simon Plouffe_
%E A000028 Entry revised by _N. J. A. Sloane_, Dec 20 2007, restoring the original definition, correcting the entries and adding a new b-file.

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