Faulhaber's Formulas
Problem 545
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:
,
where ai's are rational coefficients that can be written as reduced fractions pi/qi (if ai = 0, we shall consider qi = 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,
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.
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).