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A165514
The complement of the trapezoidal numbers.
2
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
Chris Jones and Nick Lord, Characterizing Non-Trapezoidal Numbers, The Mathematical Gazette, Vol. 83, No. 497, July 1999, pp. 262-263.
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[#] == {} &]
KEYWORD
nonn
AUTHOR
Ant King, Sep 23 2009
EXTENSIONS
More terms from Amiram Eldar, Aug 12 2019
STATUS
approved