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 A057716 The non-powers of 2. 21

%I

%S 0,3,5,6,7,9,10,11,12,13,14,15,17,18,19,20,21,22,23,24,25,26,27,28,29,

%T 30,31,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,

%U 54,55,56,57,58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,74

%N The non-powers of 2.

%C a(n) is the length signature of a string plus its length.

%C The positive members of this sequence are exactly the numbers that can be expressed as the sum of two or more consecutive positive integers (cf. A138591). - _David Wasserman_, Jan 24 2002

%C Starting at 3, these are the positions of the check bits in the single-error-correcting Hamming code.

%C Except for the offset 0, sequence corresponds to numbers with at least an odd divisor. (For largest odd divisor see A000265.) - _Lekraj Beedassy_, Apr 12 2005

%C These are exactly the numbers n with the property that, given the n(n-1)/2 sums of pairs, the original numbers can be recovered uniquely. [Nick Reingold, see Winkler reference.]

%C Subsequence of A158581; A000120(a(n)) > 1. [From _Reinhard Zumkeller_, Apr 16 2009]

%C Range of A140977. [From _Reinhard Zumkeller_, Aug 15 2010]

%C A209229(a(n)) = 0. [_Reinhard Zumkeller_, Mar 07 2012]

%C A001227(a(n)) > 1. [_Reinhard Zumkeller_, May 01 2012]

%C Numbers that can be expressed as the sum of at least two integers; numbers that can be expressed as the difference of two nonconsecutive triangular numbers. - _Charles R Greathouse IV_, Jul 27 2012

%D Carlton Gamer, David W. Roeder, and John J. Watkins, Trapezoidal numbers, Mathematics Magazine 58:2 (1985), pp. 108-110.

%D P. Winkler, Mathematical Mind-Benders, Peters, Wellesley, MA, 2007; see p. 27.

%H R. Zumkeller, <a href="/A057716/b057716.txt">Table of n, a(n) for n = 0..10000</a>

%H M. A. Nyblom, <a href="http://www.fq.math.ca/Scanned/39-3/nyblom.pdf">On the representation of the integers as a difference of nonconsecutive triangular numbers</a>, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

%F a(n) = n + [log_2(n + [log_2(n)])] gives this sequence with the exception of a(1) = 1. - David W. Wilson, Mar 29 2005

%F Find k such that 2^k - (k + 1) <= n < 2^(k+1) - (k + 2), then a(n) = n + k + 1.

%F Numbers n=2a(k)-1 k>0 are such that sum_{k=0...n}B_kM(n-k)binomial(n, k)=0 where B_k is the k-th Bernoulli number and M_k the k-th Motzkin number - _Benoit Cloitre_, Oct 19 2005

%t Module[{nn=100,maxpwr},maxpwr=Floor[Log[2,nn]];Complement[Range[0,nn], 2^Range[0,maxpwr]]] (* _Harvey P. Dale_, May 24 2012 *)

%o a057716 n = a057716_list !! n

%o a057716_list = filter ((== 0) . a209229) [0..]

%o -- _Reinhard Zumkeller_, Mar 07 2012

%o (PARI) print1(0);for(n=1,5,for(m=2^n+1,2^(n+1)-1,print1(", "m))) \\ _Charles R Greathouse IV_, Mar 07, 2012

%Y Complement of A000079. Cf. A057717, A001227, A138591.

%Y See A074894 for more about the question of when the sums of n numbers taken k at a time determine the numbers.

%K nonn,easy,changed

%O 0,2

%A John Lindgren (john.lindgren(AT)Eng.Sun.COM), Oct 24 2000

%E Better description from Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 29 2001

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