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%I #31 Apr 29 2023 14:12:11
%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
%C It appears that these are also the numbers k with the property that all noncentral widths of the symmetric representation of sigma(k) are 1's, with a(1) = 1. _Omar E. Pol_, Mar 04 2023
%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 Jim Smith, <a href="https://www.jstor.org/stable/30215335">Trapezoidal numbers</a>, Mathematics in School, Vol. 26, No. 5 (Nov., 1997), pp. 46-47.
%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, A237593.
%K nonn
%O 1,2
%A _Ant King_, Sep 23 2009
%E More terms from _Amiram Eldar_, Aug 12 2019
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