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.

Showing 1-4 of 4 results.

A024364 Ordered perimeters of primitive Pythagorean triangles.

Original entry on oeis.org

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

Views

Author

David W. Wilson

Keywords

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

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).

A103606 Primitive Pythagorean triples in nondecreasing order of perimeter, with each triple in increasing order, and if perimeters coincide then increasing order of the even members.

Original entry on oeis.org

3, 4, 5, 5, 12, 13, 8, 15, 17, 7, 24, 25, 20, 21, 29, 12, 35, 37, 9, 40, 41, 28, 45, 53, 11, 60, 61, 16, 63, 65, 33, 56, 65, 48, 55, 73, 13, 84, 85, 36, 77, 85, 39, 80, 89, 20, 99, 101, 65, 72, 97
Offset: 1

Views

Author

Alexandre Wajnberg, Mar 24 2005

Keywords

Comments

The NAME was corrected by a proposal of Wolfdieter Lang. - Ralf Steiner, Sep 29 2019
The corresponding perimeters are given in A024364. - Wolfdieter Lang, Oct 06 2014
Note that the multiplicity of primitive Pythagorean triples (increasingly ordered) with perimeter P is not always 1. See A024408 for P numbers with multiplicity k >= 2, and the first example with k = 2 for P = 1716. - Wolfdieter Lang, Sep 24 2019

References

  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 34, page 328.

Links

Crossrefs

Subsequence of A103605.
Cf. A024364, A024408.

Programs

  • Mathematica
    A103605 = Cases[Import["https://oeis.org/A103605/b103605.txt", "Table"], {, }][[All, 2]];
SortBy[Select[Partition[A103605, 3], GCD @@ # == 1&], {#[[1]] + #[[2]] + #[[3]]&, If[EvenQ[#[[1]]], #[[1]], #[[2]]]&}] // Flatten (* Jean-François Alcover, May 26 2020 *)

Extensions

Corrected at the suggestion of Ralf Steiner by Wolfdieter Lang, Sep 24 2019
Errors in b-file noticed by Kevin Ryde corrected by Jean-François Alcover, May 26 2020

A350039 Perimeters of more than one primitive 60-degree integer triangle.

Original entry on oeis.org

1260, 2520, 2574, 3080, 3740, 3780, 3978, 4620, 4940, 5148, 5720, 5814, 5940, 6435, 6930, 7020, 7280, 7560, 7820, 7866, 7956, 8190, 8550, 8580, 8892, 9044, 10010, 10350, 10395, 10472, 10640, 11628, 11880, 12006, 12240, 12870, 12920, 13050, 13260, 13340, 13680, 13685, 13832, 13860, 13950
Offset: 1

Views

Author

Seiichi Manyama, Dec 10 2021

Keywords

Examples

			399^2 + 440^2 - 399*440 = 421^2, 56^2 + 615^2 - 56*615 = 589^2 and 399 + 440 + 421 = 56 + 615 + 589 = 1260. So 1260 is a term.
5159^2 + 5904^2 - 5159*5904 = 5569^2, 3344^2 + 7119^2 - 3344*7119 = 6169^2, 1287^2 + 7952^2 - 1287*7952 = 7393^2 and 5159 + 5904 + 5569 = 3344 + 7119 + 6169 = 1287 + 7952 + 7393 = 16632. So 16632 is a term.

Links

Crossrefs

Cf. A024408, A242039, A350038.

Programs

  • Ruby
    def A(n)
  ary = []
  (1..n).each{|i|
    (i + 1..n).each{|j|
      if i.gcd(j) == 1 && (i - j) % 3 > 0
        x, y, z = j * j, i * j, i * i
        ary << 2 * x + 5 * y + 2 * z
        ary << 3 * x + 3 * y
      end
    }
  }
  ary
end
p A(100).group_by(&:to_i).select{|k, v| v.size > 1}.keys.sort[0..50]

A179271 Odd long legs `B` of more than one primitive Pythagorean triangle.

Original entry on oeis.org

2145, 3315, 3465, 4095, 4845, 5005, 5865, 6435, 6545, 6555, 7735, 8645, 9009, 9945, 10005, 10695, 11305, 11781, 13167, 13485, 13685, 13923, 14535, 15015, 15295, 15561, 16065, 16095, 17017, 17205, 17255, 17835, 17955, 18837, 19019, 19065
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 06 2010

Keywords

Comments

2145,752,2273;2145,1568,2657;;3315,812,3413;3315,2852,4373;

Links

Crossrefs

Cf. A024407, A024408, A024409, A024410, A024411, A161962, A174830

Programs

  • Mathematica
    lst1={};lst={0};Do[Do[If[GCD[a,b]==1,c=Sqrt[a^2+b^2];If[IntegerQ[c],AppendTo[lst,b];L=Length[lst];If[lst[[L]]==lst[[L-1]]&&OddQ[lst[[L]]],Print[lst[[L]]];AppendTo[lst1,lst[[L]]]]]],{a,b-1,3,-1}],{b,4,4*7!}];lst1

Extensions

a(20) - a(36) Robert G. Wilson v, Jul 12 2010
Showing 1-4 of 4 results.

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