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A024364 Ordered perimeters of primitive Pythagorean triangles.

%I A024364 #65 Jul 09 2025 17:06:47
%S A024364 12,30,40,56,70,84,90,126,132,144,154,176,182,198,208,220,234,240,260,
%T A024364 286,306,312,330,340,374,380,390,408,418,420,442,456,462,476,494,510,
%U A024364 532,544,546,552,570,598,608,644,646,650,672,684,690,700,714,736,756
%N A024364 Ordered perimeters of primitive Pythagorean triangles.
%C A024364 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives perimeters A+B+C.
%C A024364 k is in this sequence iff A070109(k) > 0. This is a subsequence of A010814.
%C A024364 For the corresponding primitive Pythagorean triples see A103606. - _Wolfdieter Lang_, Oct 06 2014
%C A024364 Any term in this sequence can be generated by f(m,k) = 2*m*(m+k), where m and k are positive coprime integers and m > 1, k < m, and m and k are not both odd. For example: f(2,1) = 2*2*(2+1) = 12. - _Agola Kisira Odero_, Apr 29 2016
%H A024364 Ray Chandler, <a href="/A024364/b024364.txt">Table of n, a(n) for n = 1..10000</a> (duplicates removed by Sean A. Irvine)
%H A024364 Leon Bernstein, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/27-1/bernstein.pdf">On primitive Pythagorean triangles with equal perimeters</a>, The Fibonacci Quarterly 27.1 (1989) 2-6 (and the earlier Bernstein paper 20.3 (1982) 227-241, see A024408).
%H A024364 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>
%F A024364 a(n) = 2*A020886(n).
%p A024364 isA024364 := proc(an) local r::integer,s::integer ; for r from floor((an/4)^(1/2)) to floor((an/2)^(1/2)) do for s from r-1 to 1 by -2 do if 2*r*(r+s) = an and gcd(r,s) < 2 then RETURN(true) ; fi ; if 2*r*(r+s) < an then break ; fi ; od ; od : RETURN(false) ; end : for n from 2 to 400 do if isA024364(n) then printf("%d,",n) ; fi ; od ; # _R. J. Mathar_, Jun 08 2006
%t A024364 isA024364[an_] := Module[{r, s}, For[r = Floor[(an/4)^(1/2)], r <= Floor[(an/2)^(1/2)], r++, For[s = r - 1, s >= 1, s -= 2, If[2r(r + s) == an && GCD[r, s] < 2, Return[True]]; If[2r(r + s) < an, Break[]]]]; Return[False]];
%t A024364 Select[Range[2, 1000], isA024364] (* _Jean-François Alcover_, May 24 2024, after _R. J. Mathar_ *)
%o A024364 (PARI) select( {is_A024364(n)=my(k=valuation(n,2), o=n>>k); k && fordiv(o, r, r^2<<(k-1) >= o && return; r^2<<k > o && gcd(r,o/r)==1 && return(1))}, [1..400]*2) \\ _M. F. Hasler_, Jul 08 2025
%Y A024364 Cf. A020886 (semiperimeters: a(n)/2), A024408 (terms with multiplicity > 1).
%K A024364 nonn
%O A024364 1,1
%A A024364 _David W. Wilson_