A331228 -id:A331228 - cp's OEIS frontend

A331676 Ceiling of circumradius of triangle with consecutive prime sides.

5, 8, 7, 9, 10, 12, 15, 17, 20, 22, 24, 26, 28, 32, 34, 37, 39, 41, 44, 46, 49, 53, 56, 58, 60, 62, 64, 68, 72, 77, 79, 82, 85, 89, 91, 94, 97, 101, 103, 107, 109, 112, 114, 117, 123, 128, 131, 133, 135, 138, 141, 145, 149, 152, 155, 158, 160, 162, 166, 171, 176, 180

Offset: 2

Frank M Jackson, Jan 24 2020

nonn

The sequence starts at offset 2 because using the first three primes yields a triangle with sides (2,3,5) that is degenerate with infinite circumradius.

Also the first two triangles in this sequence with sides (3,5,7) and (5,7,11) are obtuse and do not have their circumcentres within the bounds of the triangle. Thereafter, the triangles are acute and their circumcentres lie within the bounds of the triangle.

			a(2)=5 because a triangle with sides 3,5,7 has area = (1/4)*sqrt((3+5+7)(3+5-7)(3-5+7)(-3+5+7)) = 6.495... and circumradius = 3*5*7/(4A) = 4.041...

Harvey P. Dale, Table of n, a(n) for n = 2..1000
Eric Weisstein's World of Mathematics, Circumradius.

Cf. A096377, A330096, A331227, A331228.

lst = {}; Do[{a, b, c}={Prime[n], Prime[n+1], Prime[n+2]}; s=(a+b+c)/2; A=Sqrt[s(s-a)(s-b)(s-c)]; R=a*b*c/(4 A); AppendTo[lst, Ceiling@R], {n, 2, 200}]; lst
ccr[{a_,b_,c_}]:=Module[{s=(a+b+c)/2,A},A=Sqrt[s(s-a)(s-b)(s-c)];Ceiling[(a*b*c)/(4A)]]; ccr/@Partition[Prime[Range[2,70]],3,1] (* Harvey P. Dale, Aug 02 2025 *)

Circumradius R of a triangle with sides a, b, c is given by R = a*b*c/(4A) where the area A is given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) and where s = (a+b+c)/2.

A331224 Numerator of squared radius of circumscribed circle of a triangle with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. Denominators are A331225.

Original entry on oeis.org

1, 64, 49, 1024, 2025, 4096, 25600, 2401, 7744, 148225, 8281, 2073600, 123904, 774400, 3705625

Offset: 1

Hugo Pfoertner, Jan 14 2020

			Correspondence of the first terms b(n) = a(n)/A331225(n) with triangles (i, j, k):
b(1) = 1/3: (1,1,1), start with 1 = A331226(1) triangle.
b(2) = 64/15: (2,3,4), (2,4,4) is the first occurrence of 2 = A331226(2) triangles with identical R.
b(3) = 49/3: (3,5,7), (3,7,8), (5,7,8), (7,7,7) is the first occurrence of more triangles with identical R than the previous record 2, new record is 4 = A331226(3).
b(4) = 1024/15: (5,8,12), (5,14,16), (8,8,14), (8,12,16), (8,16,16), (12,14,16) is the first occurrence of more triangles with identical R than the previous record 4, new record is 6 = A331226(4).

Cf. A331040, A331041, A331225, A331226, A331227, A331228.

Squared radius of circumcircle of triangle with sides a, b, c:

R^2 = (a*b*c)^2 / (16*s*(s - a)*(s - b)*(s - c)) with s = (a + b + c)/2.

A331225 Denominator of squared radius of circumscribed circle of a triangle with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. Numerators are A331224.

Original entry on oeis.org

3, 15, 3, 15, 11, 15, 39, 3, 7, 96, 3, 119, 7, 39, 96

Offset: 1

Hugo Pfoertner, Jan 14 2020

			See A331224.

Cf. A331040, A331041, A331224, A331226, A331227, A331228.

A331226 a(n) is the number of triangles with integer sides i <= j <= k with squared radius of circumscribed circle b(n) = A331224(n)/A331225(n). Records of numbers of distinct triangles such that all smaller radii produce fewer triangles sharing the same radius of circumcircle than the current radius b(n).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 13, 17, 22, 31, 33, 46, 53, 67

Offset: 1

Hugo Pfoertner, Jan 14 2020

			See A331224.

Cf. A331040, A331041, A331043, A331224, A331225, A331227, A331228.

A331241 Integer values that can be assumed by the squared radius of the circumcircle of a triangle with integer sides.

Original entry on oeis.org

3, 12, 25, 27, 48, 75, 100, 108, 112, 147, 162, 169, 192, 225, 240, 243, 289, 300, 363, 400, 405, 432, 448, 507, 588, 625, 648, 675, 676, 768, 841, 867, 891, 900, 960, 972

Offset: 1

Hugo Pfoertner, Jan 13 2020

Values of A331227(k) at positions k for which A331228(k) = 1.

			a(1) = 3, because the triangle with sides (3,3,3) is the first triangle for which R^2 has an integer value.
a(2) = 12: (6,6,6),
a(3) = 25: (6,8,10) scaled-up Pythagorean triangle (3,4,5),
a(4) = 27: (9,9,9),
a(5) = 48: (12,12,12),
...
a(10) = 147: (9,15,21).

Cf. A331227, A331228.

Squared radius of circumcircle of triangle with sides a, b, c:

R^2 = (a*b*c)^2 / (16*s*(s - a)*(s - b)*(s - c)) with s = (a + b + c)/2.

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A331676 Ceiling of circumradius of triangle with consecutive prime sides.

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A331224 Numerator of squared radius of circumscribed circle of a triangle with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. Denominators are A331225.

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A331225 Denominator of squared radius of circumscribed circle of a triangle with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. Numerators are A331224.

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A331241 Integer values that can be assumed by the squared radius of the circumcircle of a triangle with integer sides.

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