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%I #31 Feb 28 2024 15:13:57
%S 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,
%T 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,
%U 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
%N Primitive triples for integer-sided triangles whose sides a < b < c are in arithmetic progression.
%C 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.
%C When b is prime, all the corresponding triples in A336750 are primitive triples.
%C The only right integer triangle in the data corresponds to the triple (3, 4, 5).
%C The number of primitive such triangles whose middle side = b is equal to A023022(b) for b >= 3.
%C For all the triples (primitive or not), miscellaneous properties and references, see A336750.
%H Neal Gersh Tolunsky, <a href="/A336755/b336755.txt">Table of n, a(n) for n = 1..10000</a>
%e The table begins:
%e 2, 3, 4;
%e 3, 4, 5;
%e 3, 5, 7;
%e 4, 5, 6;
%e 5, 6, 7;
%e 4, 7, 10;
%e 5, 7, 9;
%e 6, 7, 8;
%e The smallest such primitive triple is (2, 3, 4).
%e The only triangle with perimeter = 12 corresponds to the Pythagorean triple: (3, 4, 5).
%e There exist two triangles with perimeter = 15 corresponding to triples (3, 5, 7) and (4, 5, 6).
%e There exists only one primitive triangle with perimeter = 18 whose triple is (5, 6, 7), because (4, 6, 8) is not a primitive triple.
%p for b from 3 to 20 do
%p for a from b-floor((b-1)/2) to b-1 do
%p c := 2*b - a;
%p if gcd(a,b)=1 and gcd(b,c)=1 then print(a,b,c); end if;
%p end do;
%p end do;
%t 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 *)
%o (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
%Y Cf. A336750 (triples, primitive or not), this sequence (primitive triples), A336756 (perimeter of primitive triangles), A336757 (number of such primitive triangles whose perimeter = n).
%Y Cf. A103606 (similar for primitive Pythagorean triples).
%Y Cf. A023022.
%K nonn,tabf
%O 1,1
%A _Bernard Schott_, Sep 07 2020
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