# Problem 545: Faulhaber's Formulas
The sum of the kth powers of the first n positive integers can be
expressed as a polynomial of degree k+1 with rational coefficients, the
Faulhaber's Formulas: \$1\^k + 2\^k + ... + n\^k = \\sum\_{i=1}\^n i\^k
= \\sum\_{i=1}\^{k+1} a\_{i} n\^i = a\_{1} n + a\_{2} n\^2 + ... +
a\_{k} n\^k + a\_{k+1} n\^{k + 1}\$, where ai's are rational
coefficients that can be written as reduced fractions pi/qi (if ai = 0,
we shall consider qi = 1). For example, \$1\^4 + 2\^4 + ... + n\^4 =
-\\frac 1 {30} n + \\frac 1 3 n\^3 + \\frac 1 2 n\^4 + \\frac 1 5
n\^5.\$ Define D(k) as the value of q1 for the sum of kth powers (i.e.
the denominator of the reduced fraction a1). Define F(m) as the mth
value of k ≥ 1 for which D(k) = 20010. You are given D(4) = 30 (since
a1 = -1/30), D(308) = 20010, F(1) = 308, F(10) = 96404. Find F(105).