cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A263728 Primitive Pythagorean triples (a, b, c) in lexicographic order, with a < b < c.

Original entry on oeis.org

3, 4, 5, 5, 12, 13, 7, 24, 25, 8, 15, 17, 9, 40, 41, 11, 60, 61, 12, 35, 37, 13, 84, 85, 15, 112, 113, 16, 63, 65, 17, 144, 145, 19, 180, 181, 20, 21, 29, 20, 99, 101, 21, 220, 221, 23, 264, 265, 24, 143, 145, 25, 312, 313, 27, 364, 365, 28, 45, 53
Offset: 1

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Author

Colin Barker, Nov 20 2015

Keywords

Comments

a(3*k+1)*a(3*k+2) / (a(3*k+1)+a(3*k+2)+a(3*k+3)) is always an integer for k >= 0. Also note that a(3*k+1)*a(3*k+2)/2 is never a perfect square. - Altug Alkan, Apr 08 2016

Examples

			The first few triples are [3, 4, 5], [5, 12, 13], [7, 24, 25], [8, 15, 17], [9, 40, 41], [11, 60, 61], [12, 35, 37], [13, 84, 85], [15, 112, 113], [16, 63, 65], [17, 144, 145], [19, 180, 181], [20, 21, 29], [20, 99, 101], ... - _N. J. A. Sloane_, Dec 15 2015
		

References

  • H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, Chapter 5, Section 5.3.

Crossrefs

Programs

  • Maple
    a:=[]; b:={}; M:=30;
    for u from 2 to M do for v from 1 to u-1 do
       if gcd(u,v)=1 and u+v mod 2 = 1 then t1:=u^2-v^2; t2:= 2*u*v; t3:=u^2+v^2;
       w:=sort([t1,t2]); a:=[op(a), [op(w),t3]]; b:={ op(b), op(w), t3};
       fi:
    od: od:
    a;
    sort(a); # A263728
    sort(b); # A016825 and A042965 (Maple code from N. J. A. Sloane, Dec 15 2015)
  • PARI
    \\ Primitive Pythagorean triples (a,b,c) with a