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 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. (Formerly M0520 N0187) 17

%I M0520 N0187

%S 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 54,56,59,60,61,66,67,70,71,72,73,78,79,81,83,84,88,89,90,96,97,101,

%U 102,103,104,105,107,108,109,110,113,114,121,126,127,128,130,131,132,135,136,137

%N 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 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 Contains (for example) 180, so is different from A123193. - Max Alekseyev, Sep 20 2007

%D J. Lambek and L. Moser, On some two way classifications of integers, Canad. Math. Bull. 2 (1959), 85-89.

%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="/A000028/b000028.txt">Table of n, a(n) for n = 1..10000</a>

%p (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 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 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 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 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 [Mathematica code that implements the definition]

%t Rest[Select[Range[150],OddQ[Count[Flatten[IntegerDigits[#,2]&/@ Transpose[ FactorInteger[#]][[2]]],1]]&]] (* From Harvey P. Dale, Feb 25 2012 *)

%o a000028 n = a000028_list !! (n-1)

%o a000028_list = filter (odd . sum . map a000120 . a124010_row) [1..]

%o -- _Reinhard Zumkeller_, Oct 05 2011

%Y Cf. A133008, A000379 (complement), A000120 (binary weight function), A064547; also A066724, A026477, A050376, A084400.

%Y Note that A000069 and A001969, also A000201 and A001950 give other decompositions of the integers into two classes.

%Y Cf. A124010 (prime exponents).

%K nonn,nice,easy

%O 1,1

%A _N. J. A. Sloane_, _Simon Plouffe_

%E 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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