A263728 -id:A263728 - cp's OEIS frontend

A264826 Primitive Eisenstein triples: (a,b,c) in lexicographic order such that a^2 + b^2 - a*b - c^2 = 0, a < b < c, and gcd(a, b) = 1.

3, 7, 8, 5, 7, 8, 5, 19, 21, 7, 13, 15, 7, 37, 40, 8, 13, 15, 9, 61, 65, 11, 31, 35, 11, 91, 96, 13, 43, 48, 13, 127, 133, 15, 169, 176, 16, 19, 21, 16, 49, 55, 17, 73, 80, 17, 217, 225, 19, 91, 99, 19, 271, 280, 21, 331, 341, 23, 133, 143, 23, 397, 408

Offset: 1

Colin Barker, Nov 26 2015

The sides of a primitive 60-degree integer triangle.

Colin Barker, Table of n, a(n) for n = 1..9999
Eric Weisstein's World of Mathematics, Pythagorean Triple
Wikipedia, Eisenstein triple

Cf. A089025, A121992, A201223, A263728, A264827.

pt60(a) = {
  my(L=List(), n=-3*a^2, f, g, b, c);
  fordiv(n, f,
    g=n\f;
    if(f>g && (g+f)%2==0 && (f-g)%4==0,
      b=(f-g)\4; c=((f+g)\2+a)\2;
      if(c>0 && a

A357274 List of primitive triples for integer-sided triangles with angles A < B < C and C = 2*Pi/3 = 120 degrees.

Original entry on oeis.org

3, 5, 7, 7, 8, 13, 5, 16, 19, 11, 24, 31, 7, 33, 37, 13, 35, 43, 16, 39, 49, 9, 56, 61, 32, 45, 67, 17, 63, 73, 40, 51, 79, 11, 85, 91, 19, 80, 91, 55, 57, 97, 40, 77, 103, 24, 95, 109, 13, 120, 127, 23, 120, 133, 65, 88, 133, 69, 91, 139, 56, 115, 151, 25, 143, 157, 75, 112, 163, 15, 161, 169, 104, 105, 181

Offset: 1

Bernard Schott, Sep 22 2022

The only triangles with integer sides that have an angle equal to a whole number of degrees are triangles which have an angle of 60° (A335893), or an angle of 90° (A263728) or an angle of 120° as here (see Keith Selkirk link, p. 251).
The triples are displayed in nondecreasing order of largest side c, and if largest sides coincide then by increasing order of the smallest side a, hence, each triple (a, b, c) is in increasing order.
The corresponding metric relation between sides is c^2 = a^2 + a*b + b^2.
The triples (a, b, c) can be generated with integers u, v such that gcd(u,v) = 1 and 0 < v < u:
-> a = u^2 - v^2
-> b = 2*u*v + v^2
-> c = u^2 + u*v + v^2.
Note that side c cannot be even when the triple is primitive as here.
The (3, 5, 7) triangle is the only primitive triangle with a 120-degree angle and with its integer sides in arithmetic progression (A336750). This smallest triple is obtained for u = 2 and v = 1.
The Fermat point of these triangles is vertex C, then distance FA+FB+FC = CA+CB = b+a is an integer.
If (a,b,c) is a primitive 120-triple, then both (a,a+b,c) and (a+b,b,c) are 60-triples in A335893, see Emrys Read link, lemma 2 p. 302.

			Table of triples begins:
   3,  5,  7;
   7,  8, 13;
   5, 16, 19;
  11, 24, 31;
   7, 33, 37;
............
(7, 8, 13) is a triple for this sequence because from the law of cosines (see link), cos(C) = (7^2 + 8^2 - 13^2)/(2*7*8) = -1/2.

G. Julia, Triangles dont un angle mesure 120 degrés, Problème Capes (in French).
Emrys Read, On integer-sided triangles containing angles of 120° or 60°, Mathematical Gazette 90, July 2006, 299-305.
Keith Selkirk, Integer-Sided Triangles with an Angle of 120°, Mathematical Gazette, Vol. 67, No. 442 (Dec., 1983), pp. 251-255.
Eric Weisstein's World of Mathematics, Law of Cosines.
Wikipedia, Triangles with an angle of 120°, Eisenstein triple.

Cf. A357275, A357276, A357277, A357278.

Cf. also A263728, A336750, A335893 (similar with an angle of Pi/3).

for c from 5 to 181 by 2 do
for a from 3 to c-2 do
b := (-a + sqrt(4*c^2-3*a^2))/2;
if b=floor(b) and gcd(a,b)=1 and a

Extensions

a(31..33) = 40,51,79 inserted by Georg Fischer, Dec 04 2022

A375463 Numbers appearing on all three positions in ordered primitive Pythagorean triples.

Original entry on oeis.org

221, 325, 377, 425, 493, 629, 697, 725, 925, 1025, 1073, 1189, 1325, 1517, 1537, 1769, 1885, 1961, 2173, 2257, 2405, 2501, 2665, 2701, 2993, 3145, 3233, 3293, 3445, 3485, 3649, 3869, 3965, 3977, 4453, 4505, 4717, 4745, 5141, 5185, 5353, 5429, 5777, 5785, 5917

Offset: 1

Piotr Lipski, Aug 16 2024

nonn

			221 is a term since the following primitive Pythagorean triples have 221 in first, second and third position: (221, 24420, 24421), (60, 221, 229), (21, 220, 221).

Mathematics Stack Exchange, Is there an integer that serves as the short leg of a primitive Pythagorean triple, the long leg of another, and the hypotenuse of a third?
Rémy Sigrist, PARI program

Intersection of A008846, A024352 and A024354.

Cf. A263728.

PARI
```
\\ See Links section.
```

a(n) == 1 (mod 4). - Hugo Pfoertner, Aug 18 2024

More terms from Rémy Sigrist, Aug 17 2024

A349120 Primitive Pythagorean triples [a, b, c] in lexicographic order with a < b < c such that [w(a), w(b), w(c)] is also a primitive Pythagorean triple, where w(n) is the binary weight of n.

Original entry on oeis.org

11, 60, 61, 19, 180, 181, 25, 312, 313, 35, 612, 613, 41, 840, 841, 47, 1104, 1105, 49, 1200, 1201, 52, 165, 173, 57, 176, 185, 67, 2244, 2245, 97, 4704, 4705, 104, 153, 185, 105, 208, 233, 105, 608, 617, 131, 8580, 8581, 133, 156, 205, 145, 408, 433, 145, 10512, 10513, 165, 532, 557, 181, 16380, 16381, 193, 18624, 18625

Offset: 1

Ctibor O. Zizka, Nov 08 2021

			[11, 60, 61] is a primitive Pythagorean triple, and [w(11), w(60), w(61)] = [3, 4, 5] is also a primitive Pythagorean triple, thus 11, 60, and 61 are members.

Cf. A000120, A263728.

ppt(a) = {my(L=List(), b, c, d, g); fordiv(a^2, d, g=a^2\d; if(d<=g && (d+g)%2==0, c=(d+g)\2; b=g-c; if(aA263728
isok(t) = {my(ht = vecsort(apply(hammingweight, t))); (ht[1]^2 + ht[2]^2 == ht[3]^2) && (gcd(ht)==1);}
lista(nn) = {my(list=List()); for (n=1, nn, my(v = ppt(n)); if (#v, for (k=1, #v, if (isok(v[k]), listput(list, v[k]));););); Vec(list);} \\ Michel Marcus, Nov 10 2021

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A264826 Primitive Eisenstein triples: (a,b,c) in lexicographic order such that a^2 + b^2 - a*b - c^2 = 0, a < b < c, and gcd(a, b) = 1.

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A357274 List of primitive triples for integer-sided triangles with angles A < B < C and C = 2*Pi/3 = 120 degrees.

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Maple

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A375463 Numbers appearing on all three positions in ordered primitive Pythagorean triples.

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PARI

Formula

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A349120 Primitive Pythagorean triples [a, b, c] in lexicographic order with a < b < c such that [w(a), w(b), w(c)] is also a primitive Pythagorean triple, where w(n) is the binary weight of n.

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PARI