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A336756 Perimeters in increasing order of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression. 4
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..69.

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;

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

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Last modified April 17 11:26 EDT 2021. Contains 343064 sequences. (Running on oeis4.)