A165513 Trapezoidal numbers.

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

Offset: 1

Author

Ant King, Sep 23 2009

Keywords

• easy
• nonn

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).

Examples

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

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.

Crossrefs

Cf. A138591, A165514, A019434.

Programs

Mathematica

Trapezoidal[n_]:=Module[{result},result={};Do[sum=0;start=i;lis={};m=i; While[sum0 &]