

A057716


The nonpowers of 2.


30



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

0,2


COMMENTS

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 singleerrorcorrecting 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(n1)/2 sums of pairs, the original numbers can be recovered uniquely. [Nick Reingold, see Winkler reference.]
Subsequence of A158581; A000120(a(n)) > 1.  Reinhard Zumkeller, Apr 16 2009
Range of A140977.  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 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*k1) divides binomial(2*k1,k). See Ihringer & Kupavskii.  Michel Marcus, Oct 02 2017


REFERENCES

C. Ballantine, M. Merca, Padovan numbers as sums over partitions into odd parts, Journal of Inequalities and Applications, (2016) 2016:1. doi:10.1186/s1366001509525
J. M. R. Caballero, A characterization of the hypotenuses of primitive Pythagorean triangles ..., Amer. Math. Monthly 126 (2019), 7477.
Martin Davis, "Algorithms, Equations, and Logic", pp. 415 of S. Barry Cooper and Andrew Hodges, Eds., "The Once and Future Turing: Computing the World", Cambridge 2016.
P. Winkler, Mathematical MindBenders, Peters, Wellesley, MA, 2007; see p. 27.


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Carlton Gamer, David W. Roeder, and John J. Watkins, Trapezoidal numbers, Mathematics Magazine 58:2 (1985), pp. 108110.
Ferdinand Ihringer, Andrey Kupavskii, Regular Intersecting Families, arXiv:1709.10462 [math.CO], 2017. See Lemma 24 p. 11.
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256263.
Henri Picciotto, Staircases


FORMULA

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(nk)*binomial(n, k) = 0 where B_k is the kth Bernoulli number and M_k the kth Motzkin number.  Benoit Cloitre, Oct 19 2005
From Robert Israel, May 05 2015: (Start)
G.f.: (1x)^(2)*Sum(m>=0, x^(2^mm)*(2^m*x2^m*x^2+x) + x^(2^(m+1)m)*(2^(m+1)*x2^(m+1)x)).
a(im) = i for 2^m < i < 2^(m+1).
a(n) = A103586(n) + n for n >= 1. (End)


MAPLE

select(t > t/2^padic:ordp(t, 2) <> 1, [$0..100]); # Robert Israel, May 05 2015


MATHEMATICA

Module[{nn = 100, maxpwr}, maxpwr = Floor[Log[2, nn]]; Complement[Range[0, nn], 2^Range[0, maxpwr]]] (* Harvey P. Dale, May 24 2012 *)
Complement[Range[0, 99], 2^Range[0, 7]] (* Alonso del Arte, May 05 2015 *)


PROG

(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


CROSSREFS

Complement of A000079. Cf. A057717, A001227, A103586, A138591, A138592.
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: A079581 A229858 A269020 * A138591 A136492 A062506
Adjacent sequences: A057713 A057714 A057715 * A057717 A057718 A057719


KEYWORD

nonn,easy


AUTHOR

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


EXTENSIONS

Better description from Ahmed Fares (ahmedfares(AT)mydeja.com), Apr 29 2001


STATUS

approved



