A103605 -id:A103605 - cp's OEIS frontend

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.

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

Alexandre Wajnberg, Mar 24 2005

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

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

Jean-François Alcover, Table of n, a(n) for n = 1..3975
Ron Knott, The Pythagorean triples.
Michael Penn, Number Theory | Primitive Pythagorean Triples, Youtube video, 2019.

Subsequence of A103605.

Cf. A024364, A024408.

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 *)

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

A336750 Triples for integer-sided triangles whose sides a < b < c are in arithmetic progression.

Original entry on oeis.org

2, 3, 4, 3, 4, 5, 3, 5, 7, 4, 5, 6, 4, 6, 8, 5, 6, 7, 4, 7, 10, 5, 7, 9, 6, 7, 8, 5, 8, 11, 6, 8, 10, 7, 8, 9, 5, 9, 13, 6, 9, 12, 7, 9, 11, 8, 9, 10, 6, 10, 14, 7, 10, 13, 8, 10, 12, 9, 10, 11, 6, 11, 16, 7, 11, 15, 8, 11, 14, 9, 11, 13, 10, 11, 12, 7, 12, 17, 8, 12, 16

Offset: 1

Bernard Schott, Aug 03 2020

The triples are displayed in increasing order of perimeter, and if perimeters coincide then by increasing order of the smallest side, hence, each triple (a, b, c) is in increasing order.
Equivalently: triples of integer-sided triangles such that b = (a+c)/2 with a < c.
As the perimeter of these triangles = 3*b, the triples are also displayed in increasing order of middle side.
When a < b < c are in arithmetic progression with b - a = c - b = x, then 1 <= x <= floor((b-1)/2), hence, there exist for each side b >= 3, floor((b-1)/2) = A004526(b) triangles whose sides a < b < c are in arithmetic progression.
The only right integer-sided triangles such that a < b < c are in arithmetic progression correspond to the Pythagorean triples (3k, 4k, 5k) with k > 0.
There do not exist triangles whose sides a < b < c and angles A < B < C are both in arithmetic progression.
Three geometrical properties about these triangles, even if they are not integer-sided:
   1) tan(A/2) * tan(C/2) = 1/3,
   2) r = h_b/3, where r is the inradius and h_b the length of the altitude through B,
   3) The line (IG) is parallel to side (AC), where I is the incenter and G is the centroid of the triangle.

			The smallest such triangle is (2, 3, 4).
The only triangle with perimeter = 12 corresponds to the Pythagorean triple: (3, 4, 5).
There exist two triangles with perimeter = 15 corresponding to triples (3, 5, 7) and (4, 5, 6).
There exist also two triangles with perimeter = 18 corresponding to triples (4, 6, 8) and (5, 6, 7).
The table begins:
  2, 3, 4;
  3, 4, 5;
  3, 5, 7;
  4, 5, 6;
  4, 6, 8;
  5, 6, 7;
  4, 7, 10;
  5, 7, 9;
  6, 7, 8;

V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-288 p. 120, André Desvigne.

Michel Marcus, Table of n, a(n) for n = 1..16650 (Rows 1 to 5550)
Wikipedia, Integer triangle.

Cf. A336751 (smallest side), A307136 (middle side), A336753 (largest side), A336754 (perimeter), A024164 (number of triangles with perimeter = n), A336755 (primitive triples), A336756 (perimeter of primitive triangles), A336757 (number of primitive triangles with perimeter = n).
Cf. A004526 (number of triangles with middle side = b).
Cf. A103605 (similar, with Pythagorean triples).
Cf. A335893 (similar, with A, B, C in arithmetic progression).

for b from 3 to 20 do
for a from b-floor((b-1)/2) to b-1 do
c := 2*b - a;
print(a,b,c);
end do;
end do;

Block[{nn = 12, a, b, c}, Reap[Do[Do[Sow@ {a, b, 2 b - a}, {a, b - Floor[(b - 1)/2], b - 1}], {b, 3, nn}]][[-1, 1]] ] // Flatten (* Michael De Vlieger, Oct 15 2020 *)

tabf(nn) = {for (b = 3, nn, for (a = b-floor((b-1)/2), b-1, my(c = 2*b - a); print(a, " ", b, " ", c);););} \\ Michel Marcus, Sep 08 2020

T(n,1) = A336751(n); T(n,2) = A307136(n); T(n,3) = A336753(n).

A336754(n) = T(n,1) + T(n,2) + T(n,3).

A009096 Ordered perimeters of Pythagorean triangles, listed with multiplicity.

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 56, 60, 60, 70, 72, 80, 84, 84, 90, 90, 96, 108, 112, 120, 120, 120, 126, 132, 132, 140, 144, 144, 150, 154, 156, 160, 168, 168, 168, 176, 180, 180, 180, 182, 192, 198, 200, 204, 208, 210, 210, 216, 220, 224, 228, 234, 240, 240, 240, 240, 252, 252

Offset: 1

David W. Wilson

nonn

			The perimeters are listed with multiplicity, for example a(8) = a(9) = 60 = 15 + 20 + 25 = 10 + 24 + 26. - _M. F. Hasler_, Jul 04 2025

Cf. A103605 (the corresponding triples), A024364 (perimeters of primitive Pythagorean triangles).

a(n) = A103605(3n) + A103605(3n-1) + A103605(3n-2). - M. F. Hasler, Jul 04 2025

Name clarified by M. F. Hasler, Jul 04 2025

A347594 a(0) = 1; for n>0, a(n) is the smallest positive integer such that a(n-1)^2 + n^2 + a(n) is a square.

Original entry on oeis.org

1, 2, 1, 6, 12, 27, 19, 31, 64, 48, 96, 72, 1, 26, 28, 15, 3, 26, 24, 24, 48, 64, 44, 35, 48, 96, 108, 151, 131, 223, 447, 831, 639, 190, 380, 299, 507, 663, 1212, 904, 209, 7, 36, 104, 17, 87, 116, 211, 264, 264, 165, 103, 143, 151, 204, 303, 536, 1055, 860, 1688, 3156, 2592, 1341, 1399

Offset: 0

Scott R. Shannon, Sep 08 2021

In the first one million terms the largest value is a(987016) = 123592518669. In this range the smallest number that has not yet appeared is 9.

			a(1) = 2 as a(0)^2 + 1^2 = 1 + 1 = 2, and 2 + 2 = 4 = 2^2 is the next smallest square.
a(2) = 1 as a(1)^2 + 2^2 = 4 + 4 = 8, and 8 + 1 = 9 = 3^2 is the next smallest square.
a(60) = 3156 as a(59)^2 + 60^2 = 2849344 + 3600 = 2852944, and 2852944 + 3156 = 2856100 = 1690^2 is the next smallest square.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000290, A103605, A009000, A005408, A347754.

Nest[Append[#, Block[{k = 1, m = Last[#1]}, While[! IntegerQ@ Sqrt[#2^2 + m^2 + k], k++]; k]] & @@ {#, Length@ #} &, {1}, 63] (* Michael De Vlieger, Sep 08 2021 *)

lista(nn) = {my(prec = 1, list=List()); listput(list, prec); for (n=1, nn, my(k = 1); while (!issquare(prec^2+n^2+k), k++); listput(list, k); prec = k;); Vec(list);} \\ Michel Marcus, Sep 13 2021

from math import isqrt
A347594_list = [1]
for n in range(1,10**3):
    m = A347594_list[n-1]**2+n**2
    A347594_list.append((isqrt(m)+1)**2-m) # Chai Wah Wu, Sep 12 2021

A332978 The number of regions formed inside a triangle with leg lengths equal to the Pythagorean triples by straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

Original entry on oeis.org

271, 5746, 14040, 32294, 50551, 108737, 180662, 276533, 259805, 558256, 591687, 901811, 1117126, 1015277, 1386667, 1223260, 1944396, 3149291, 3165147, 4523784, 4764416, 4859839, 6025266, 7186096

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 04 2020

The terms are from numeric computation - no formula for a(n) is currently known.

			The triples are ordered by the total sum of the leg lengths:
           Triple        |      Number of regions
          (3, 4, 5)      |           271
          (6, 8, 10)     |           5746
          (5, 12, 13)    |           14040
          (9, 12, 15)    |           32294
          (8, 15, 17)    |           50551
          (12, 16, 20)   |           108737
          (7, 24, 25)    |           180662
          (15, 20, 25)   |           276533
          (10, 24, 26)   |           259805
          (20, 21, 29)   |           558256
          (18, 24, 30)   |           591687
          (16, 30, 34)   |           901811
          (21, 28, 35)   |           1117126
          (12, 35, 37)   |           1015277
          (15, 36, 39)   |           1386667
          (9, 40, 41)    |           1223260
          (24, 32, 40)   |           1944396
          (27, 36, 45)   |           3149291
          (14, 48, 50)   |           3165147
          (20, 48, 52)   |           4523784
          (24, 45, 51)   |           4764416
          (30, 40, 50)   |           4859839
          (28, 45, 53)   |           6025266
          (33, 44, 55)   |           7186096

Scott R. Shannon, Triangle regions for leg lengths (3,4,5).
Scott R. Shannon, Triangle regions for leg lengths (6,8,10).
Scott R. Shannon, Triangle regions for leg lengths (5,12,13).
Scott R. Shannon, Triangle regions for leg lengths (9,12,15).
Scott R. Shannon, Triangle regions for leg lengths (8,15,17).
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Wikipedia, Pythagorean triple.

Cf. A333135 (n-gons), A333136 (vertices), A333137 (edges), A103605 (Pythagorean triple ordering), A007678, A092867, A331452.

a(8)-a(24) from Lars Blomberg, Jun 07 2020

A333135 Irregular table read by rows: Take a triangle with Pythagorean triple leg lengths with all diagonals drawn, as in A332978. Then T(n,k) = number of k-sided polygons in that figure for k >= 3 where the legs are divided into unit length parts.

Original entry on oeis.org

139, 94, 34, 3, 1, 2383, 2421, 760, 167, 13, 2, 5307, 5958, 2113, 563, 80, 17, 2, 13083, 13560, 4479, 1002, 153, 16, 1, 18827, 20896, 8256, 2139, 377, 49, 6, 1, 42992, 45400, 15930, 3771, 579, 60, 5, 63526, 79275, 28922, 7315, 1404, 202, 14, 4

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

See A332978 for the Pythagorean triple ordering and the links for images of the triangles.

			Table begins:
139, 94, 34, 3, 1;
2383, 2421, 760, 167, 13, 2;
5307, 5958, 2113, 563, 80, 17, 2;
13083, 13560, 4479, 1002, 153, 16, 1;
18827, 20896, 8256, 2139, 377, 49, 6, 1;
42992, 45400, 15930, 3771, 579, 60, 5;
63526, 79275, 28922, 7315, 1404, 202, 14, 4;
The row sums are A332978.

Lars Blomberg, Table of n, a(n) for n = 1..213 (the first 24 rows)

Cf. A332978 (regions), A333136 (vertices), A333137 (edges), A103605 (Pythagorean triple ordering), A007678, A092867, A331452.

Corrected typo in a(12) and a(49) and beyond from Lars Blomberg, Jun 07 2020

A333136 The number of vertices formed on a triangle with leg lengths equal to the Pythagorean triples by the straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

Original entry on oeis.org

230, 5138, 13181, 29277, 48107, 100003, 173261, 256910, 247940, 541752, 554717, 869197, 1051503, 987045, 1333241, 1190131, 1843049, 2991447, 3073340, 4382249, 4630456, 4635744, 5914142, 6877208

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

See A332978 for the Pythagorean triple ordering and the links for images of the triangles.

Cf. A332978 (regions), A333135 (n-gons), A333137 (edges), A103605 (Pythagorean triple ordering), A092866, A332599, A007569.

a(8)-a(24) from Lars Blomberg, Jun 07 2020

A333137 The number of edges formed on a triangle with leg lengths equal to the Pythagorean triples by the straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

Original entry on oeis.org

500, 10883, 27220, 61570, 98657, 208739, 353922, 533442, 507744, 1100007, 1146403, 1771007, 2168628, 2002321, 2719907, 2413390, 3787444, 6140737, 6238486, 8906032, 9394871, 9495582, 11939407, 14063303

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

See A332978 for the Pythagorean triple ordering and the links for images of the triangles.

Cf. A332978 (regions), A333135 (n-gons), A333136 (vertices), A103605 (Pythagorean triple ordering), A274586 , A332600, A331765.

a(8)-a(24) from Lars Blomberg, Jun 07 2020

A105520 Sums of area and perimeter of Pythagorean triples, sorted in increasing order, including duplicates.

Original entry on oeis.org

18, 48, 60, 90, 100, 140, 144, 180, 210, 270, 280, 288, 294, 320, 360, 378, 448, 462, 480, 594, 600, 648, 660, 720, 728, 756, 858, 900, 900, 924, 980, 1008, 1008, 1078, 1080, 1120, 1170, 1210, 1260, 1344, 1496, 1530, 1530, 1568, 1584, 1584, 1680, 1700, 1728

Offset: 1

Alexandre Wajnberg, May 02 2005

			a(28) = 900 = (18+80+82) + (18*80/2) for 18*18 + 80*80 = 82*82.
a(29) = 900 = (25+60+65) + (25*60/2) for 25*25 + 60*60 = 65*65.
a(32) = 1008 = (24+70+74) + (24*70/2) for 24*24 + 70*70 = 74*74.
a(33) = 1008 = (36+48+60) + (36*48/2) for 36*36 + 48*48 = 60*60.

Giovanni Resta, Table of n, a(n) for n = 1..10711 (terms < 10^7)
Douglas Butler, Pythagorean triples [including multiples] up to 2100, Oundle School iCT Training Centre.
Ron Knott, Pythagorean triples.

Cf. A103605, A103606, A024364, A024406, A105521.

L = {}; mx = 1728; Do[ Do[ If[ IntegerQ[z = Sqrt[x^2 + y^2]], v = x y/2 + x + y + z; If[v <= mx, AppendTo[L, v], Break[]]], {y, x-1}], {x, 4, 4 + (2 mx^2)^(1/3)}]; Sort@ L (* Giovanni Resta, Mar 16 2020 *)

T. = 0                        ;  S = ''
do C = 1 to 999               ;  H = C*C
   do D = 1 to C              ;  I = D*D
      do E = 1 to D           ;  J = E*E
         if I + J < H   then  iterate E
         if I + J = H   then  do
            K = T.0 + 1       ;  T.0 = K
            P = C + D + E     ;  A = ( D * E ) / 2
            T.K = right( A + P, 6 )
            T.K = T.K '=' A '+' P '(' E '+' D '+' C ')'
         end
         leave E
      end E
   end D
end C
call KWIK 'T.' /* sort by A+P for area A and perimeter P */
Y = 0
do N = 1 to T.0 while length( S ) < 255
   X = word( T.N, 1 )         ;  say T.N
   if X <= Y   then  say 'dupe:' N - 1 N ':' Y X
   S = S || ', ' || X         ;  Y = X
end N
say substr( S, 3 )            /* Frank Ellermann, Mar 02 2020 */

Corrected and extended by Frank Ellermann, Mar 02 2020

A334382 Least k whose set of divisors contains exactly n Pythagorean triples, or 0 if no such k exists.

Original entry on oeis.org

60, 120, 240, 360, 960, 720, 3840, 1440, 2160, 2880, 8160, 3600, 69360, 8400, 8640, 7200, 32640, 9360, 16800, 14400, 34560, 24480, 130560, 18720, 77760, 54600, 28080, 25200, 67200, 37440, 11045580, 61200, 73440, 97920, 294000, 46800, 65520, 50400, 268800, 109200

Offset: 1

Michel Lagneau, Apr 26 2020

This is a subsequence of A169823: a(n) == 0 (mod 60) because one side of every Pythagorean triple is divisible by 3, another by 4, and another by 5. The smallest and best-known Pythagorean triple is (a, b, c) = (3, 4, 5).

			a(3) = 240 because the set of divisors {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240} contains 3 Pythagorean triples: (3, 4, 5), (6, 8, 10) and (12, 16, 20). The first triple is primitive.

Giovanni Resta, Table of n, a(n) for n = 1..450

Cf. A103605, A103606, A169823, A334080.

with(numtheory):
for n from 1 to 52 do :
ii:=0:
for k from 60 by 60 to 10^8 while(ii=0) do:
   d:=divisors(k):n0:=nops(d):it:=0:
    for i from 1 to n0-1 do:
     for j from i+1 to n0-2 do :
      for m from i+2 to n0 do:
       if d[i]^2 + d[j]^2 = d[m]^2
        then
        it:=it+1:
        else
       fi:
      od:
     od:
    od:
    if it = n
     then
     ii:=1: printf (`%d %d \n`,n,k):
     else
    fi:
od:
od:

a(31) from Giovanni Resta, Apr 27 2020

cp's OEIS Frontend

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.

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A336750 Triples for integer-sided triangles whose sides a < b < c are in arithmetic progression.

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A009096 Ordered perimeters of Pythagorean triangles, listed with multiplicity.

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A347594 a(0) = 1; for n>0, a(n) is the smallest positive integer such that a(n-1)^2 + n^2 + a(n) is a square.

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A332978 The number of regions formed inside a triangle with leg lengths equal to the Pythagorean triples by straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

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A333135 Irregular table read by rows: Take a triangle with Pythagorean triple leg lengths with all diagonals drawn, as in A332978. Then T(n,k) = number of k-sided polygons in that figure for k >= 3 where the legs are divided into unit length parts.

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A333136 The number of vertices formed on a triangle with leg lengths equal to the Pythagorean triples by the straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

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Extensions

A333137 The number of edges formed on a triangle with leg lengths equal to the Pythagorean triples by the straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

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A105520 Sums of area and perimeter of Pythagorean triples, sorted in increasing order, including duplicates.