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A057716 The nonpowers of 2.

%I A057716 #108 May 25 2025 09:25:06
%S A057716 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 A057716 30,31,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,
%U A057716 54,55,56,57,58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,74
%N A057716 The nonpowers of 2.
%C A057716 a(n) is the length signature of a string plus its length.
%C A057716 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 A057716 Starting at 3, these are the positions of the data bits in the single-error-correcting Hamming code.
%C A057716 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
%C A057716 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 A057716 Subsequence of A158581; A000120(a(n)) > 1. - _Reinhard Zumkeller_, Apr 16 2009
%C A057716 Range of A140977. - _Reinhard Zumkeller_, Aug 15 2010
%C A057716 A209229(a(n)) = 0. - _Reinhard Zumkeller_, Mar 07 2012
%C A057716 A001227(a(n)) > 1. - _Reinhard Zumkeller_, May 01 2012
%C A057716 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
%C A057716 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
%D A057716 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.
%D A057716 Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 67-69.
%D A057716 P. Winkler, Mathematical Mind-Benders, Peters, Wellesley, MA, 2007; see p. 27.
%H A057716 Reinhard Zumkeller, <a href="/A057716/b057716.txt">Table of n, a(n) for n = 0..10000</a>
%H A057716 C. Ballantine and M. Merca, <a href="https://www.researchgate.net/publication/289250007_Padovan_numbers_as_sums_over_partitions_into_odd_parts">Padovan numbers as sums over partitions into odd parts</a>, Journal of Inequalities and Applications, (2016) 2016:1; <a href="https://doi.org/10.1186/s13660-015-0952-5">doi</a>.
%H A057716 Carlton Gamer, David W. Roeder, and John J. Watkins, <a href="http://www.maa.org/sites/default/files/Carlton_Gamer,_David_W03006._Roeder_and_John_J._Watkins.pdf">Trapezoidal numbers</a>, Mathematics Magazine 58:2 (1985), pp. 108-110.
%H A057716 Ferdinand Ihringer and Andrey Kupavskii, <a href="https://arxiv.org/abs/1709.10462">Regular Intersecting Families</a>, arXiv:1709.10462 [math.CO], 2017. See Lemma 24 p. 11.
%H A057716 M. A. Nyblom, <a href="https://web.archive.org/web/2024*/https://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.
%H A057716 Henri Picciotto, <a href="http://www.mathedpage.org/teachers/staircases.pdf">Staircases</a>.
%H A057716 J. M. Rodriguez Caballero, <a href="https://www.jstor.org/stable/48661362">A characterization of the hypotenuses of primitive pythagorean triangles using partitions into consecutive parts</a>, Amer. Math. Monthly 126 (2019), 74-77.
%F A057716 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 A057716 Find k such that 2^k - (k + 1) <= n < 2^(k+1) - (k + 2), then a(n) = n + k + 1.
%F A057716 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
%F A057716 From _Robert Israel_, May 05 2015: (Start)
%F A057716 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)).
%F A057716 a(i-m) = i for 2^m < i < 2^(m+1).
%F A057716 a(n) = A103586(n) + n for n >= 1. (End)
%p A057716 select(t -> t/2^padic:-ordp(t,2) <> 1, [$0..100]); # _Robert Israel_, May 05 2015
%t A057716 Module[{nn = 100,maxpwr},maxpwr = Floor[Log[2, nn]]; Complement[Range[0, nn], 2^Range[0, maxpwr]]]  (* _Harvey P. Dale_, May 24 2012 *)
%t A057716 Complement[Range[0, 99], 2^Range[0, 7]] (* _Alonso del Arte_, May 05 2015 *)
%o A057716 (Haskell)
%o A057716 a057716 n = a057716_list !! n
%o A057716 a057716_list = filter ((== 0) . a209229) [0..]
%o A057716 -- _Reinhard Zumkeller_, Mar 07 2012
%o A057716 (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
%o A057716 (Python)
%o A057716 def A057716(n): return n + (n + n.bit_length()).bit_length() # _Matthew Andres Moreno_, Jun 16 2024
%o A057716 (Python)
%o A057716 from itertools import count, islice
%o A057716 def agen(): # generator of terms
%o A057716     yield 0
%o A057716     yield from (j for i in count(0) for j in range(2**i+1, 2**(i+1)))
%o A057716 print(list(islice(agen(), 70))) # _Michael S. Branicky_, Oct 11 2024
%Y A057716 Complement of A000079. Cf. A057717, A001227, A103586, A138591, A138592.
%Y A057716 See A074894 for more about the question of when the sums of n numbers taken k at a time determine the numbers.
%K A057716 nonn,easy
%O A057716 0,2
%A A057716 John Lindgren (john.lindgren(AT)Eng.Sun.COM), Oct 24 2000
%E A057716 Better description from Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 29 2001