Solution:

For positive $x, y$ with $xy \lt 1,$ the tangent addition formula gives $\arctan x + \arctan y = \arctan\frac{x + y}{1 - xy}.$ Applying it twice: $$\arctan\frac{1}{3} + \arctan\frac{1}{4} = \arctan\frac{\frac{1}{3} + \frac{1}{4}}{1 - \frac{1}{12}} = \arctan\frac{7}{11}, \qquad \arctan\frac{7}{11} + \arctan\frac{1}{5} = \arctan\frac{\frac{7}{11} + \frac{1}{5}}{1 - \frac{7}{55}} = \arctan\frac{23}{24}.$$

The equation becomes $\arctan\frac{23}{24} + \arctan\frac{1}{n} = \arctan 1,$ so $\frac{23/24 + 1/n}{1 - 23/(24n)} = 1.$ Clearing denominators, $23n + 24 = 24n - 23,$ giving $n = 47.$