A336755 Primitive triples for integer-sided triangles whose sides a < b < c are in arithmetic progression.

2, 3, 4, 3, 4, 5, 3, 5, 7, 4, 5, 6, 5, 6, 7, 4, 7, 10, 5, 7, 9, 6, 7, 8, 5, 8, 11, 7, 8, 9, 5, 9, 13, 7, 9, 11, 8, 9, 10, 7, 10, 13, 9, 10, 11, 6, 11, 16, 7, 11, 15, 8, 11, 14, 9, 11, 13, 10, 11, 12, 7, 12, 17, 11, 12, 13, 7, 13, 19, 8, 13, 18, 9, 13, 17, 10, 13, 16, 11, 13, 15, 12, 13, 14

Offset

1,1

Comments

The triples are displayed in increasing order of perimeter (equivalently in increasing order of middle side) and if perimeters coincide then by increasing order of the smallest side; also, each triple (a, b, c) is in increasing order.

When b is prime, all the corresponding triples in A336750 are primitive triples.

The only right integer triangle in the data corresponds to the triple (3, 4, 5).

The number of primitive such triangles whose middle side = b is equal to A023022(b) for b >= 3.

For all the triples (primitive or not), miscellaneous properties and references, see A336750.

Links

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Example

The table begins:
  2, 3, 4;
  3, 4, 5;
  3, 5, 7;
  4, 5, 6;
  5, 6, 7;
  4, 7, 10;
  5, 7, 9;
  6, 7, 8;
The smallest such primitive triple 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 exists only one primitive triangle with perimeter = 18 whose triple is (5, 6, 7), because (4, 6, 8) is not a primitive triple.

Maple

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

Mathematica

Select[Flatten[Table[{a, b, 2*b-a}, {b, 3, 20}, {a, b-Floor[(b-1)/2], b-1}], 1], GCD @@ # == 1 &] (* Paolo Xausa, Feb 28 2024 *)

Prog

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

Crossrefs

Cf. A336750 (triples, primitive or not), this sequence (primitive triples), A336756 (perimeter of primitive triangles), A336757 (number of such primitive triangles whose perimeter = n).

Cf. A103606 (similar for primitive Pythagorean triples).

Cf. A023022.

Sequence in context: A221108 A205554 A336750 * A214613 A364801 A325933

Adjacent sequences: A336752 A336753 A336754 * A336756 A336757 A336758

Keyword

nonn,tabf

Author

Bernard Schott, Sep 07 2020

Status

approved