

A165514


The complement of the trapezoidal numbers.


1



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
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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^(k1)*(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 nonneighboring 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


REFERENCES

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


LINKS

Table of n, a(n) for n=1..40.
Chris Jones and Nick Lord, Characterizing NonTrapezoidal Numbers, The Mathematical Gazette, Vol. 83, No. 497, July 1999, pp. 262263.
Ron Knott, Introducing Runsums
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.
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



