|
%I #18 Jun 11 2023 11:46:33
%S 5,7,9,11,12,13,14,15,17,18,19,20,21,22,23,24,25,26,27,29,30,31,33,34,
%T 35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,
%U 58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81
%N Trapezoidal numbers.
%C Trapezoidal numbers are polite numbers (A138591) that have a runsum representation which excludes one, and hence that can be depicted graphically by a trapezoid. Jones and Lord have shown that this is the sequence of integers excluding the powers of 2, the perfect numbers and 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).
%D Smith, Jim: Trapezoidal numbers, Mathematics in School (November 1997).
%H Peter Kagey, <a href="/A165513/b165513.txt">Table of n, a(n) for n = 1..10000</a>
%H Chris Jones and Nick Lord, <a href="http://www.jstor.org/stable/3619053">Characterizing Non-Trapezoidal Numbers</a>, The Mathematical Gazette, Vol. 83, No. 497, July 1999, pp. 262-263.
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/runsums/index.html#calc">Introducing runsums</a>
%H Melvyn B. Nathanson, <a href="http://arxiv.org/abs/1601.07058">Trapezoidal numbers, divisor functions, and a partition theorem of Sylvester</a>, arXiv:1601.07058 [math.NT], 2016.
%H T. Verhoeff, <a href="https://cs.uwaterloo.ca/journals/JIS/trapzoid.html">Rectangular and Trapezoidal Arrangements</a>, J. Integer Sequences, Vol. 2, 1999, #99.1.6.
%e As 12=3+4+5 is the fifth integer with a runsum representation which excludes one, then a(5)=12.
%t 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[100],Length[Trapezoidal[ # ]]>0 &]
%Y Cf. A138591, A165514, A019434.
%K easy,nonn
%O 1,1
%A _Ant King_, Sep 23 2009
|
