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

1,1

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

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

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A000005, A068068.

Sequence in context: A306231 A125703 A349381 * A348243 A019567 A098286

Adjacent sequences: A156685 A156686 A156687 * A156689 A156690 A156691

Keyword

easy,nice,nonn

Author

Ant King, Feb 18 2009

Extensions

More terms from Antti Karttunen, Sep 27 2018

Status

approved