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

1,2

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

Links

Table of n, a(n) for n=1..40.

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.

Example

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

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[#] == {} &]

Crossrefs

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

Sequence in context: A066816 A247334 A237450 * A182417 A189704 A223543

Adjacent sequences: A165511 A165512 A165513 * A165515 A165516 A165517

Keyword

nonn

Author

Ant King, Sep 23 2009

Extensions

More terms from Amiram Eldar, Aug 12 2019

Status

approved