A165514 The complement of the trapezoidal numbers.

1, 2, 3, 4, 6, 8, 10, 16, 28, 32, 64, 128, 136, 256, 496, 512, 1024, 2048, 4096, 8128, 8192, 16384, 32768, 32896, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33550336, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824

Offset: 1

Author

Ant King, Sep 23 2009

Keywords

• nonn

Comments

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).

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

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

Examples

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

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..2577
• Chris Jones and Nick Lord, Characterizing Non-Trapezoidal Numbers, The Mathematical Gazette, Vol. 83, No. 497, July 1999, pp. 262-263.
• Ron Knott, Introducing Runsums
• Jim Smith, Trapezoidal numbers, Mathematics in School, Vol. 26, No. 5 (Nov., 1997), pp. 46-47.
• T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

Crossrefs

Cf. A138591, A165513, A019434, A000396, A000079, A000217, A002808, A237593.

Programs

Mathematica

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[#] == {} &]

Extensions

More terms from Amiram Eldar, Aug 12 2019