Search: id:A000028
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%I A000028 M0520 N0187
%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,
               54,
%T A000028 56,59,60,61,66,67,70,71,72,73,78,79,81,83,84,88,89,90,96,97,101,102,103,
%U A000028 104,105,107,108,109,110,113,114,121,126,127,128,130,131,132,135,136,137
%N A000028 Let n = p_1^e_1 p_2^e_2 p_3^e_3 ... be the prime factorization of n. 
               Sequence gives n 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
%D A000028 J. Lambek and L. Moser, On some two way classifications of integers, 
               Canad. Math. Bull. 2 (1959), 85-89.
%D A000028 J. 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="b000028.txt">Table of n, a(n) for n = 1..10000</
               a>
%p A000028 (Maple program from N. J. A. Sloane (njas(AT)research.att.com), 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 (wouter.meeussen(AT)pandora.be), Dec 21 2007 [Mathematica 
               code that implements the definition]
%Y A000028 Cf. A133008, A000379 (complement), A000120 (binary weight function), 
               A064547; also A066724, A026477, A050376, A084400.
%Y A000028 Note that A000069 and A001969, also A000201 and A001950 give other decompositions 
               of the integers into two classes.
%Y A000028 Sequence in context: A130520 A005706 A064175 this_sequence A026416 A123193 
               A066724
%Y A000028 Adjacent sequences: A000025 A000026 A000027 this_sequence A000029 A000030 
               A000031
%K A000028 nonn,nice,easy
%O A000028 1,1
%A A000028 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A000028 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Dec 20 2007, 
               restoring the original definition, correcting the entries and adding 
               a new b-file.

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