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A057716
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The non-powers of 2.
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22
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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, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74
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OFFSET
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0,2
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COMMENTS
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a(n) is the length signature of a string plus its length.
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
Starting at 3, these are the positions of the check bits in the single-error-correcting Hamming code.
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
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.]
Subsequence of A158581; A000120(a(n)) > 1. [From Reinhard Zumkeller, Apr 16 2009]
Range of A140977. [From Reinhard Zumkeller, Aug 15 2010]
A209229(a(n)) = 0. [Reinhard Zumkeller, Mar 07 2012]
A001227(a(n)) > 1. [Reinhard Zumkeller, May 01 2012]
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
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REFERENCES
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Carlton Gamer, David W. Roeder, and John J. Watkins, Trapezoidal numbers, Mathematics Magazine 58:2 (1985), pp. 108-110.
P. Winkler, Mathematical Mind-Benders, Peters, Wellesley, MA, 2007; see p. 27.
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LINKS
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R. Zumkeller, Table of n, a(n) for n = 0..10000
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.
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FORMULA
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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
Find k such that 2^k - (k + 1) <= n < 2^(k+1) - (k + 2), then a(n) = n + k + 1.
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
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MATHEMATICA
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Module[{nn=100, maxpwr}, maxpwr=Floor[Log[2, nn]]; Complement[Range[0, nn], 2^Range[0, maxpwr]]] (* Harvey P. Dale, May 24 2012 *)
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PROG
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(Haskell)
a057716 n = a057716_list !! n
a057716_list = filter ((== 0) . a209229) [0..]
-- Reinhard Zumkeller, Mar 07 2012
(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
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CROSSREFS
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Complement of A000079. Cf. A057717, A001227, A138591.
See A074894 for more about the question of when the sums of n numbers taken k at a time determine the numbers.
Sequence in context: A010906 A114309 A079581 * A138591 A136492 A062506
Adjacent sequences: A057713 A057714 A057715 * A057717 A057718 A057719
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KEYWORD
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nonn,easy
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AUTHOR
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John Lindgren (john.lindgren(AT)Eng.Sun.COM), Oct 24 2000
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EXTENSIONS
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Better description from Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 29 2001
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STATUS
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approved
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