A024364 Ordered perimeters of primitive Pythagorean triangles.

12, 30, 40, 56, 70, 84, 90, 126, 132, 144, 154, 176, 182, 198, 208, 220, 234, 240, 260, 286, 306, 312, 330, 340, 374, 380, 390, 408, 418, 420, 442, 456, 462, 476, 494, 510, 532, 544, 546, 552, 570, 598, 608, 644, 646, 650, 672, 684, 690, 700, 714, 736, 756

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives perimeters A+B+C.

k is in this sequence iff A070109(k) > 0. This is a subsequence of A010814.

For the corresponding primitive Pythagorean triples see A103606. - Wolfdieter Lang, Oct 06 2014

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

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (duplicates removed by Sean A. Irvine)
• Leon Bernstein, On primitive Pythagorean triangles with equal perimeters, The Fibonacci Quarterly 27.1 (1989) 2-6 (and the earlier Bernstein paper 20.3 (1982) 227-241, see A024408).
• Ron Knott, Pythagorean Triples and Online Calculators

Crossrefs

Cf. A020886 (semiperimeters: a(n)/2), A024408 (terms with multiplicity > 1).

Programs

Maple

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

Mathematica

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]];
Select[Range[2, 1000], isA024364] (* Jean-François Alcover, May 24 2024, after R. J. Mathar *)

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< o && gcd(r,o/r)==1 && return(1))}, [1..400]*2) \\ M. F. Hasler, Jul 08 2025

Formula

a(n) = 2*A020886(n).