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A352315
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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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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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a(8) inserted by and a(12)-a(34) from Jinyuan Wang, Mar 15 2022
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STATUS
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approved
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