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 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 (list; graph; refs; listen; history; text; internal format)
 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 REFERENCES Smith, Jim: Trapezoidal numbers, Mathematics in School (November 1997). LINKS 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 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, 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

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Last modified May 6 16:17 EDT 2021. Contains 343586 sequences. (Running on oeis4.)