A198453 Consider triples a<=b

2, 2, 3, 3, 5, 6, 4, 9, 10, 5, 6, 8, 5, 14, 15, 6, 9, 11, 6, 20, 21, 7, 27, 28, 8, 10, 13, 8, 35, 36, 9, 13, 16, 9, 21, 23, 9, 44, 45, 10, 26, 28, 10, 54, 55, 11, 14, 18, 11, 20, 23, 11, 65, 66, 12, 17, 21, 12, 24, 27

Offset: 1

Author

Charlie Marion, Oct 25 2011

Keywords

• nonn
• tabf

Comments

The definition can be generalized to define Pythagorean k-triples a<=b

If a, b and c form a Pythagorean k-triple, then n*a, n*b and n*c form a Pythagorean n*k-triple.

A triangle is defined to be a Pythagorean k-triangle if its sides form a Pythagorean k-triple.

If a, b and c are the sides of a Pythagorean k-triangle ABC with a<=b0 and acute when k<0. When k=0, the triangles are Pythagorean, as in the Beiler reference and Ron Knott's link.

For all k, the area of a Pythagorean k-triangle ABC with a<=b

Define a Pythagorean k-triple to be primitive if and only if there are no integers r>1, s>0 such that = for some Pythagorean s-triple . Thus, every Pythagorean 1-triple is primitive. For every k>1, the set of Pythagorean k-triples contains some non-primitive triples.

In particular, for d a proper divisor of k, it includes (k/d)*(a,b,c), where (a,b,c) is a Pythagorean d-triple. - Franklin T. Adams-Watters, Dec 01 2011

Examples

			2*3 + 2*3 = 3*4
3*4 + 5*6 = 6*7
4*5 + 9*10 = 10*11
5*6 + 6*7 = 8*9
5*6 + 14*15 = 15*16
6*7 + 9*10 = 11*12
		

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

• Ron Knott, Pythagorean Triples and Online Calculators

Crossrefs

Cf. A103606, A198454-A198469.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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