Shallit Constant

Define $f(x_1,x_2,\ldots,x_n)$ with $x_i$ Positive as

$$f(x_1, x_2, \ldots, x_n)\equiv \sum_{i=1}^n x_i+\sum_{1\leq i\leq k\leq n} \prod_{j=i}^k {1\over x_j}.$$

Then

$$\min f = 3n - C + o(1)$$

as $n$ increases, where the Shallit constant is

$$C = 1.369451403937\dots$$

(Shallit 1995). In their solution, Grosjean and De Meyer (quoted in Shallit 1995) reduced the complexity of the problem.

References

MacLeod, A. http://www.mathsoft.com/asolve/constant/shapiro/macleod.html.

Shallit, J. Solution by C. C. Grosjean and H. E. De Meyer. ``A Minimization Problem.'' Problem 94-15 in SIAM Review 37, 451-458, 1995.