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A165514 The complement of the trapezoidal numbers. 1

%I

%S 1,2,3,4,6,8,10,16,28,32,64,128,136,256,496,512,1024,2048,4096,8128,

%T 8192,16384,32768,32896,65536,131072,262144,524288,1048576,2097152,

%U 4194304,8388608,16777216,33550336,33554432,67108864,134217728,268435456,536870912,1073741824

%N The complement of the trapezoidal numbers.

%C Trapezoidal numbers (A165513) are polite numbers (A138591) that have a runsum representation which excludes one, and hence that can be depicted graphically by a trapezoid. This sequence is their complement, and Jones and Lord have shown that it is constructed from the powers of 2 (A000079), the perfect numbers (A000396) and those integers of the form 2^(k-1)*(2^k+1) where k is necessarily a power of 2 and 2^k + 1 is a Fermat prime (A019434).

%C Starting with 4, composite numbers (A002808) not a difference of non-neighboring triangular numbers (A000217). For T(x) - T(y), x - y > 1, where T are the triangular numbers, all other composite numbers can be represented as a triangular number difference. - _Ed Pegg Jr_, Feb 23 2016

%D Smith, Jim: Trapezoidal numbers, Mathematics in School (November 1997).

%H Chris Jones and Nick Lord, <a href="http://www.jstor.org/stable/3619053">Characterizing Non-Trapezoidal Numbers</a>, The Mathematical Gazette, Vol. 83, No. 497, July 1999, pp. 262-263.

%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/runsums/index.html#calc">Introducing Runsums</a>

%H T. Verhoeff, <a href="https://cs.uwaterloo.ca/journals/JIS/trapzoid.html">Rectangular and Trapezoidal Arrangements</a>, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

%e As the fifth integer which does not have a runsum representation which excludes one is 6, then a(5)=6.

%t trapezoidal[n_] := Module[{result}, result = {}; Do[sum = 0; start = i; lis = {}; m = i; While[sum < n, sum = sum + m; lis = AppendTo[lis, m]; If[sum == n, AppendTo[result, lis]]; m++], {i, 2, Floor[n/2]}]; result]; Select[Range[10000], trapezoidal[#] == {} &]

%Y Cf. A138591, A165513, A019434, A000396, A000079, A000217, A002808.

%K nonn

%O 1,2

%A _Ant King_, Sep 23 2009

%E More terms from _Amiram Eldar_, Aug 12 2019

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Last modified June 25 10:09 EDT 2021. Contains 345453 sequences. (Running on oeis4.)