A156688 The total number of distinct Pythagorean triples with an area numerically equal to n times their perimeters.

2, 3, 6, 4, 6, 9, 6, 5, 10, 9, 6, 12, 6, 9, 18, 6, 6, 15, 6, 12, 18, 9, 6, 15, 10, 9, 14, 12, 6, 27, 6, 7, 18, 9, 18, 20, 6, 9, 18, 15, 6, 27, 6, 12, 30, 9, 6, 18, 10, 15, 18, 12, 6, 21, 18, 15, 18, 9, 6, 36, 6, 9, 30, 8, 18, 27, 6, 12, 18, 27, 6, 25, 6, 9, 30, 12, 18, 27, 6, 18, 18, 9, 6, 36, 18, 9, 18, 15, 6, 45, 18, 12, 18, 9, 18

Offset: 1

Author

Ant King, Feb 18 2009

Keywords

• easy
• nice
• nonn

Comments

The members of this sequence are also 1/2 the number of divisors of 8n^2. The corresponding results for primitive triangles only are in A068068.

Also, the total number of distinct "areas with equal border", that is: Let x, y be positive integers so that the area xy equals the border around it with thickness n. As a formula it is: 2xy = (x+2n)(y+2n). To compare with the original, the areas at thickness 5 are 11x210, 12x110, 14x60, 15x50, 18x35, 20x30. - Juhani Heino, Jul 22 2012

Examples

			There are 6 Pythagorean triples whose area is 5 times their perimeters - (21,220,221), (22,120,122), (24,70,74), (25,60,65),(28,45,53) and (30,40,50) - hence a(5)=6.
		

References

• Chi, Henjin and Killgrove, Raymond; Problem 1447, Crux Math 15(5), May 1989.
• Chi, Henjin and Killgrove, Raymond; Solution to Problem 1447, Crux Math 16(7), September 1990.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Ron Knott, Right-angled Triangles and Pythagoras' Theorem

Crossrefs

Cf. A000005, A068068.

Programs

Mathematica

1/2 DivisorSigma[0,8#^2] &/@Range[75]

PARI

A156688(n) = (numdiv(8*n*n)/2); \\ Antti Karttunen, Sep 27 2018

Formula

a(n) = A000005(8n^2)/2 = A078644(2n).

Extensions

More terms from Antti Karttunen, Sep 27 2018