A120063 Shortest side c of all integer-sided triangles with sides a<=b<=c and inradius n.

5, 10, 12, 15, 25, 24, 35, 30, 36, 39, 55, 45, 65, 63, 53, 60, 85, 68, 95, 75, 77, 88, 115, 85, 125, 130, 108, 105, 145, 106, 155, 120, 132, 170, 137, 135, 185, 190, 156, 150, 205, 154, 215, 165, 159, 230, 235, 170, 245, 195, 204, 195, 265, 204, 200, 195, 228, 290

Offset: 1

Author

Hugo Pfoertner, Jun 13 2006

Keywords

• nonn

Comments

Terms a(11),..., a(100) computed by Thomas Mautsch (mautsch(AT)ethz.ch).

Empirically, 2*sqrt(3) < a(n)/n <= 5. The lower bound is provably tight, the upper bound seems to be achieved infinitely often, e.g, for prime n >= 5. It appears that a(p) = 5p for prime p != 3. - David W. Wilson, Jun 17 2006

Minimum of longest side occurring among all A120062(n) triangles having integer sides with integer inradius n.

Examples

			a(1)=5 because the only triangle with integer sides and inradius 1 is {3,4,5}; its longest side is 5.
a(2)=10: The triangles with inradius 2 are {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17}. The minimum of their longest sides is min(13,10,29,20,17)=10.
		

References

• Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Links

• David W. Wilson, Table of n, a(n) for n = 1..10000

Crossrefs

See A120062 for sequences related to integer-sided triangles with integer inradius n.

Cf. A120062 [triangles with integer inradius], A120252 [primitive triangles with integer inradius], A057721 [maximum of longest sides], A058331 [maximum of shortest sides], A120064 [minimum of middle sides], A082044 [maximum of middle sides], A005408 [minimum of shortest sides], A007237.