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A000028
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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)
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26
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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, 56, 59, 60, 61, 66, 67, 70, 71, 72, 73, 78, 79, 81, 83, 84, 88, 89, 90, 96, 97, 101, 102, 103, 104, 105, 107, 108, 109, 110, 113, 114, 121, 126, 127, 128, 130, 131, 132, 135, 136, 137
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OFFSET
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1,1
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COMMENTS
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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.
Numbers n such that infinitary Moebius function of n (A064179) equals -1. This follows from the definition of A064179.
Number n is in the sequence if and only if the number k=k(n) of terms of A050376 which divide n with odd maximal exponent is odd.
For example, if n=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.
(End)
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
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REFERENCES
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J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 22.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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If n=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
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MAPLE
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(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, ...
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
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
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
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MATHEMATICA
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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 *)
Rest[Select[Range[150], OddQ[Count[Flatten[IntegerDigits[#, 2]&/@ Transpose[ FactorInteger[#]][[2]]], 1]]&]] (* Harvey P. Dale, Feb 25 2012 *)
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PROG
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(Haskell)
a000028 n = a000028_list !! (n-1)
a000028_list = filter (odd . sum . map a000120 . a124010_row) [1..]
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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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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STATUS
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approved
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