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A352315 a(n) is the distance d between the incenter I and the circumcenter O of the integer-sided triangle whose sides correspond to the n-th primitive triple of A352314.

%I A352315 #25 Mar 15 2022 22:25:33
%S A352315 5,5,21,13,91,143,95,533,221,17,407,575,341,275,703,259,377,319,53,
%T A352315 559,4181,793,481,3599,715,784,943,1955,3965,549,7055,6815,2144,1961
%N A352315 a(n) is the distance d between the incenter I and the circumcenter O of the integer-sided triangle whose sides correspond to the n-th primitive triple of A352314.
%C A352315 The triples of sides (a, b, c) are in increasing order of largest side c.
%C A352315 For the corresponding primitive triples and miscellaneous properties, formulas and references see A352314.
%C A352315 Two distinct such triangles can have the same distance OI (see examples).
%C A352315 From the table in A352314, when d is prime and the triangle ABC isosceles, then
%C A352315 -> d divides the two equal sides of this triangle, and also,
%C A352315 -> if d^2 = R, then r = (R-1)/2,
%C A352315 -> if d^2 = 3R then r = (R-3)/2.
%e A352315 a(1) = 5 because with the smallest triple (10, 10, 16), we get s = (10+10+16)/2 = 18, A = 48, r = 48/18 = 8/3, R = (10*10*16)/(4*48) = 25/3, and d = sqrt(25/3 * 9/3) = 5 is an integer.
%e A352315 a(2) = 5 also because with the second triple (40, 40, 48), we get s = (40+40+48)/2 = 64, A = 768, r = 768/64 = 12, R = (40*40*48)/(4*768) = 25, and d = sqrt(25*(25-24)) = 5.
%e A352315 a(3) = 21 because with the third triple (16, 49, 55) that is the first triangle not isosceles, we get s = (16+49+55)/2 = 60, A = 220*sqrt(3), r = 11*sqrt(3)/3, R = (16*49*55)/(4*220*sqrt(3)) = 49*sqrt(3)/3, and d = sqrt(49^2/3 - (2*11*49)/3) = 21.
%o A352315 (PARI) lista(nn) = my(d, s); for(c=2, nn, for(b=1+c\2, c, for(a=1+c-b, b, s=(a+b+c)/2; if(denominator(d=a^2*b^2*c^2/16/s/(s-a)/(s-b)/(s-c)-a*b*c/2/s) == 1 && issquare(d) && gcd([a, b, c, d=sqrtint(d)]) == 1, print1(d, ", "))))); \\ _Jinyuan Wang_, Mar 15 2022
%Y A352315 Cf. A231174, A350378, A350379, A352314, A352316.
%K A352315 nonn,more
%O A352315 1,1
%A A352315 _Bernard Schott_, Mar 11 2022
%E A352315 a(8) inserted by and a(12)-a(34) from _Jinyuan Wang_, Mar 15 2022