A352314 -id:A352314 - cp's OEIS frontend

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.

5, 5, 21, 13, 91, 143, 95, 533, 221, 17, 407, 575, 341, 275, 703, 259, 377, 319, 53, 559, 4181, 793, 481, 3599, 715, 784, 943, 1955, 3965, 549, 7055, 6815, 2144, 1961

Offset: 1

Bernard Schott, Mar 11 2022

The triples of sides (a, b, c) are in increasing order of largest side c.
For the corresponding primitive triples and miscellaneous properties, formulas and references see A352314.
Two distinct such triangles can have the same distance OI (see examples).
From the table in A352314, when d is prime and the triangle ABC isosceles, then
-> d divides the two equal sides of this triangle, and also,
-> if d^2 = R, then r = (R-1)/2,
-> if d^2 = 3R then r = (R-3)/2.

			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.
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.
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.

Cf. A231174, A350378, A350379, A352314, A352316.

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

a(8) inserted by and a(12)-a(34) from Jinyuan Wang, Mar 15 2022

A352316 Perimeter of primitive integer-sided triangles such that the distance d = OI between the circumcenter O and the incenter I is also a positive integer.

Original entry on oeis.org

36, 128, 120, 288, 360, 500, 1008, 972, 1024, 1500, 1152, 1372, 2520, 3072, 2520, 3960, 4000, 4116, 4860, 5040, 4500, 5760, 6860, 5324, 6804, 6435, 8000, 7776, 8192, 10920, 8788, 9216, 10395, 10976

Offset: 1

Bernard Schott, Mar 14 2022

For the corresponding primitive triples, miscellaneous properties and links, see A352314.

The sequence is not increasing. For example, a(2) = 128 for triangle with largest side = 48 while a(3) = 120 for triangle with largest side = 55.

			a(1) = 36 because the smallest triple is (10, 10, 16) with corresponding d = OI = A352315(1) = 5.

Cf. A231174, A350378, A350379, A352314, A352315.

a(n) = A352314(n, 1) + A352314(n, 2) + A352314(n, 3).

a(19)-a(34) from Jinyuan Wang, Mar 14 2022

cp's OEIS Frontend

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.

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