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A165513 Trapezoidal numbers. 4
5, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

REFERENCES

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

LINKS

Peter Kagey, Table of n, a(n) for n = 1..10000

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

Melvyn B. Nathanson, Trapezoidal numbers, divisor functions, and a partition theorem of Sylvester, arXiv:1601.07058 [math.NT], 2016.

T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

EXAMPLE

As 12=3+4+5 is the fifth integer with a runsum representation which excludes one, then a(5)=12.

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[100], Length[Trapezoidal[ # ]]>0 &]

CROSSREFS

Cf. A138591, A165514, A019434.

Sequence in context: A212191 A336122 A241853 * A002342 A080353 A184108

Adjacent sequences:  A165510 A165511 A165512 * A165514 A165515 A165516

KEYWORD

easy,nonn

AUTHOR

Ant King, Sep 23 2009

STATUS

approved

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Last modified June 14 21:27 EDT 2021. Contains 345041 sequences. (Running on oeis4.)