cp's OEIS Frontend

A335895 Middle side of primitive triples, in nondecreasing order, for integer-sided triangles whose angles A < B < C are in arithmetic progression.

%I A335895 #42 Apr 04 2021 00:51:59
%S A335895 7,7,13,13,19,19,31,31,37,37,43,43,49,49,61,61,67,67,73,73,79,79,91,
%T A335895 91,91,91,97,97,103,103,109,109,127,127,133,133,133,133,139,139,151,
%U A335895 151,157,157,163,163,169,169,181,181,193,193,199,199,211,211,217,217,217,217
%N A335895 Middle side of primitive triples, in nondecreasing order, for integer-sided triangles whose angles A < B < C are in arithmetic progression.
%C A335895 Equivalently, lengths of the middle side b of primitive non-equilateral triangles that have an angle of Pi/3; indeed, this side is opposite to angle B = Pi/3.
%C A335895 Also solutions b of the Diophantine equation b^2 = a^2 - a*c + c^2 with a < b and gcd(a,b) = 1.
%C A335895 For the corresponding primitive triples and miscellaneous properties and references, see A335893.
%C A335895 When (a, b, c) is a triple or a solution, then (c-a, b, c) is another solution, so every b in the data is present an even number of times (see examples).
%C A335895 From _Bernard Schott_, Apr 02 2021: (Start)
%C A335895 Terms are primes of the form 6k+1, or products of primes of the form 6k+1. Three observations:
%C A335895 -> The lengths b are in A004611 \ {1} without repetition, 1 corresponds to the equilateral triangle (1, 1, 1).
%C A335895 -> Every term appears 2^k (k>0) times consecutively and the smallest term that appears 2^k times is precisely A121940(k); see examples.
%C A335895 -> The terms that appear precisely twice consecutively are in A133290. (End)
%F A335895 a(n) = A335893(n, 2).
%F A335895 b is such that b^2 = a^2 - a*c + c^2 with gcd(a,b) = 1 and a < b.
%e A335895 b = 7 appears twice because A121940(1) = 7 and:
%e A335895   7^2 = 3^2 - 3*8 + 8^2, with triple (3, 7, 8),
%e A335895   7^2 = 5^2 - 5*8 + 8^2, with triple (5, 7, 8).
%e A335895 b = 91 appears four times because A121940(2) = 91 and:
%e A335895   91^2 = 11^2 - 11*96 + 96^2, with triple (11, 91, 96),
%e A335895   91^2 = 85^2 - 85*96 + 96^2, with triple (85, 91, 96),
%e A335895   91^2 = 19^2 - 19*99 + 99^2, with triple (19, 91, 99),
%e A335895   91^2 = 80^2 - 80*99 + 99^2, with triple (80, 91, 99).
%e A335895 b = 1729 appears eight times because A121940(3) = 1729 and the triples of these 2^3 = 8 triangles are (96, 1729, 1775), (1679, 1729, 1775), (249, 1729, 1840), (1591, 1729, 1840), (656, 1729, 1961), (1305, 1729, 1961), (799, 1729, 1984), (1185, 1729, 1984).
%p A335895 for b from 3 to 100 by 2 do
%p A335895 for a from 1 to b-1 do
%p A335895 c := (a+ sqrt(4*b^2-3*a^2))/2;
%p A335895 if igcd(a, b) = 1 and issqr(4*b^2-3*a^2) then print(b); end if;
%p A335895 end do;
%p A335895 end do;
%o A335895 (PARI) lista(nn) = {forstep(b=1, nn, 2, for(a=1, b-1, if (gcd(a, b) == 1, my(d = 4*b^2 - 3*a^2); if (issquare(d), my(c = (a + sqrtint(d))/2); if (denominator(c)==1, print1(b, ", "));););););} \\ _Michel Marcus_, Jul 05 2020
%Y A335895 Cf. A004611, A121940, A133290.
%Y A335895 Cf. A335893 (triples), A335894 (smallest side), this sequence (middle side), A335896 (largest side), A335897 (perimeter).
%K A335895 nonn
%O A335895 1,1
%A A335895 _Bernard Schott_, Jul 04 2020