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A057716
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The nonpowers of 2.
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49
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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 data bits in the single-error-correcting Hamming code.
Except for the offset 0, sequence corresponds to numbers with at least an odd divisor > 1 (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.]
Numbers that can be expressed as the sum of at least two consecutive integers; numbers that can be expressed as the difference of two nonconsecutive triangular numbers. - Charles R Greathouse IV, Jul 27 2012
Except for the 1st term 0, these are the integers k such that 2*(2*k-1) divides binomial(2*k-1,k). See Ihringer & Kupavskii. - Michel Marcus, Oct 02 2017
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REFERENCES
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Martin Davis, "Algorithms, Equations, and Logic", pp. 4-15 of S. Barry Cooper and Andrew Hodges, Eds., "The Once and Future Turing: Computing the World", Cambridge 2016.
J. M. Rodriguez Caballero, A characterization of the hypotenuses of primitive Pythagorean triangles ..., Amer. Math. Monthly 126 (2019), 74-77.
P. Winkler, Mathematical Mind-Benders, Peters, Wellesley, MA, 2007; see p. 27.
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LINKS
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Carlton Gamer, David W. Roeder, and John J. Watkins, Trapezoidal numbers, Mathematics Magazine 58:2 (1985), pp. 108-110.
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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_k*M(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
G.f.: (1-x)^(-2)*Sum(m>=0, x^(2^m-m)*(2^m*x-2^m*x^2+x) + x^(2^(m+1)-m)*(2^(m+1)*x-2^(m+1)-x)).
a(i-m) = i for 2^m < i < 2^(m+1).
a(n) = A103586(n) + n for n >= 1. (End)
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MAPLE
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select(t -> t/2^padic:-ordp(t, 2) <> 1, [$0..100]); # Robert Israel, May 05 2015
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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..]
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CROSSREFS
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See A074894 for more about the question of when the sums of n numbers taken k at a time determine the numbers.
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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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