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

%I A336756 #16 Feb 29 2024 11:30:07
%S A336756 9,12,15,15,18,21,21,21,24,24,27,27,27,30,30,33,33,33,33,33,36,36,39,
%T A336756 39,39,39,39,39,42,42,42,45,45,45,45,48,48,48,48,51,51,51,51,51,51,51,
%U A336756 51,54,54,54,57,57,57,57,57,57,57,57,57,60,60,60,60,63,63,63,63,63,63
%N A336756 Perimeters in increasing order of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression.
%C A336756 Equivalently: perimeters of primitive integer-sided triangles such that b = (a+c)/2 with a < c.
%C A336756 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.
%C A336756 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).
%C A336756 For the corresponding primitive triples, miscellaneous properties, and references, see A336750.
%H A336756 Paolo Xausa, <a href="/A336756/b336756.txt">Table of n, a(n) for n = 1..10000</a>
%e A336756 Perimeter = 9 only for the smallest triangle (2, 3, 4).
%e A336756 Perimeter = 12 only for the Pythagorean triple (3, 4, 5).
%e A336756 Perimeter = 15 for the two triples (3, 5, 7) and (4, 5, 6).
%e A336756 There only exists one primitive triangle with perimeter = 18 whose triple is (5, 6, 7), because (4, 6, 8) is not a primitive triple.
%p A336756 for b from 3 to 21 do
%p A336756 for a from b-floor((b-1)/2) to b -1 do
%p A336756 c := 2*b - a;
%p A336756 if gcd(a,b)=1 and gcd(b,c)=1 then print(a+b+c); end if;
%p A336756 end do;
%p A336756 end do;
%t A336756 Flatten[Array[ConstantArray[3*#, EulerPhi[#]/2] &, 20, 3]] (* _Paolo Xausa_, Feb 29 2024 *)
%o A336756 (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
%Y A336756 Cf. A336754 (perimeters, primitive or not), A336755 (primitive triples), this sequence (perimeters of primitive triangles), A336757 (number of such primitive triangles whose perimeter = n).
%Y A336756 Cf. A023022.
%K A336756 nonn
%O A336756 1,1
%A A336756 _Bernard Schott_, Sep 16 2020