[seqfan] How to list pairs, triples, etc.
Richard Guy
rkg at cpsc.ucalgary.ca
Sat May 15 23:04:35 CEST 2010
Dear all,
Perhaps this is already well known to those who
well know it, but is there a way to list triples, say, in
OEIS ? Partial answer: the hypotenuses of Pythagorean
triples are listed in A009003.
Here is a sequence which seems not to be already
in, which someone may like to submit. It arises from
what might be called antipythagorean triples, namely
{a,b,c} where b+c = x^2, c+a=y^2, a+b=z^2 are each
perfect squares. Now any positive (or indeed negative)
integer can occur in an antipythagorean triple (in
fact in infinitely many, I believe), but 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.
Some numbers can be (the largest) members of more than
one antipythagorean triple, e.g., {120,24,1} {120,76,24}
or {126,99,70} {126,100,44} --- should these (largest
members) appear more than once in the sequence.
[I sh'd've said that I want my triples to consist of
distinct positive integers, for example, I don't count
(288,288,1} though some might think that I should.]
I get the first few antipythagorean triples to be
{30,19,6} {44,20,5} {47,34,2} {48,33,16} {60,21,4}
{66,34,15} {69,52,12} {70,51,30} {78,22,3} {86,35,14}
{90,54,10} {92,52,29} {94,75,6} {95,74,26} ...
so that the proposed sequence is (E&OE, as usual)
30, 44, 47, 48, 60, 66, 69, 70, 78, 86, 90, 92, 94,
95, 96, 98, 108, 113, 116, 118, 118, 120 (twice?),
122, 125, 126 (twice?), 132, 138, 142, 147, 150, 152,
154, 156, 157, 158 (twice?), 165, 170, 176, 180, 185,
186, 188, 194 (twice?), 195, 196, 197, 198, 200, 207,
212, 214, 216, 218, 221, 222, 224, 227, 230, 232, ...
Let me know of any I've missed, and if you can prove
that all sufficiently large numbers are in there
(or not!) and what is meant by ``sufficiently large''.
Thanks to anyone interested. R.
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