[seqfan] Re: How to list pairs, triples, etc.
Richard Guy
rkg at cpsc.ucalgary.ca
Mon May 17 23:03:36 CEST 2010
Hans,
This is a very good piece of work.
There may need to be another idea in
order to show that all sufficiently large
numbers occur as largest members of triples.
Meanwhile you seem to have several
likely candidates for new sequences in
OEIS.
Thankyou very much for your interest
and help. R.
On Mon, 17 May 2010, Hans Havermann wrote:
> 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}.
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