# [seqfan] Re: How to list pairs, triples, etc.

Hans Havermann pxp at rogers.com
Mon May 17 22:19:52 CEST 2010

```Richard Guy:

>                               ... I am interested in

> which positive integers can be the largest member of
> an antipythagorean triple.  Perhaps all sufficiently
> large numbers, but that's an open question as far as
> I'm presently concerned.

Consider the sequence of the number of antipythagorean-triple
solutions ({a,b,c}, a > b > c > 0) given the positive integers as
their largest member:

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1,
1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1,
0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0,
1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1,
0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 1, 0,
0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0,
1, 1, 0, 2, 0, 1, 0, 1, 0, 3, 1, 1, 1, 1, 0, 1, ...

You ask if the number of zeros finite. I think so. I have found only
2627 zeros in this sequence up to index 1.3 million, the final ten
sitting at indices 275395, 276951, 285289, 285829, 324049, 335449,
392635, 410929, 581101, and 692481. The finitude of zeros (as well as
ones, twos, threes, etc.) makes more sense in the context of the
average number of solutions increasing as we approach infinity. Yes,
there is a lot of scatter, but I trust that that scatter will fail to
reach the smaller values as we advance. The lack of a single zero from
0.7 to 1.3 million certainly bears this out.

By the way, the smallest positive integers that have exactly {0, 1,
2, ... 60} solutions are {1, 30, 120, 194, 282, 870, 1322, 1220, 1442,
2240, 3128, 3842, 3812, 5288, 5378, 6662, 7592, 8408, 6722, 10448,
10922, 12098, 10592, 15248, 17618, 16112, 18722, 20738, 21842, 26888,
29138, 26408, 20162, 28802, 27458, 36758, 30608, 44258, 44072, 33728,
48578, 43688, 52658, 48962, 49088, 53762, 63362, 55682, 55538, 55568,
73448, 74258, 72738, 68642, 62498, 83882, 81698, 81218, 92978, 92072,
96818}.

```