A336756 Perimeters in increasing order of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression.

9, 12, 15, 15, 18, 21, 21, 21, 24, 24, 27, 27, 27, 30, 30, 33, 33, 33, 33, 33, 36, 36, 39, 39, 39, 39, 39, 39, 42, 42, 42, 45, 45, 45, 45, 48, 48, 48, 48, 51, 51, 51, 51, 51, 51, 51, 51, 54, 54, 54, 57, 57, 57, 57, 57, 57, 57, 57, 57, 60, 60, 60, 60, 63, 63, 63, 63, 63, 63

Offset

1,1

Comments

Equivalently: perimeters of primitive integer-sided triangles such that b = (a+c)/2 with a < c.

As perimeter = 3 * middle side, these perimeters p are all multiples of 3 and each term p appears consecutively A023022(p/3) = phi(p/3)/2 times for p >= 9.

Remark, when the middle side is prime, then the number of primitive triangles with a perimeter p = 3*b equals phi(p/3)/2 = (b-1)/2 = (p-3)/6 and in this case, all the triangles are primitive (see A336754).

For the corresponding primitive triples, miscellaneous properties, and references, see A336750.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Example

Perimeter = 9 only for the smallest triangle (2, 3, 4).
Perimeter = 12 only for the Pythagorean triple (3, 4, 5).
Perimeter = 15 for the two triples (3, 5, 7) and (4, 5, 6).
There only exists 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 21 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

Flatten[Array[ConstantArray[3*#, EulerPhi[#]/2] &, 20, 3]] (* Paolo Xausa, Feb 29 2024 *)

Prog

(PARI) lista(nn) = {my(list=List()); for (b = 3, nn, for (a = b-floor((b-1)/2), b-1, my(c = 2*b - a); if (gcd([a, b, c]) == 1, listput(list, a+b+c); ); ); ); Vec(list); } \\ Michel Marcus, Sep 16 2020

Crossrefs

Cf. A336754 (perimeters, primitive or not), A336755 (primitive triples), this sequence (perimeters of primitive triangles), A336757 (number of such primitive triangles whose perimeter = n).

Cf. A023022.

Sequence in context: A170951 A044859 A336754 * A114306 A009188 A138299

Adjacent sequences: A336753 A336754 A336755 * A336757 A336758 A336759

Keyword

nonn

Author

Bernard Schott, Sep 16 2020

Status

approved