0%

Problem 541


Problem 541


Divisibility of Harmonic Number Denominators

The nthharmonic number Hn is defined as the sum of the multiplicative inverses of the first n positive integers, and can be written as a reduced fraction an/bn.
$H_n = \displaystyle \sum_{k=1}^n \frac{1}{k} = \frac {a_n} {b_n}$, with$\text {gcd}(a_n, b_n)=1$.

Let M(p) be the largest value of n such that bn is not divisible by p.

For example, M(3) = 68 because H_{68} = \frac {a_{68}} {b_{68}} = \frac {14094018321907827923954201611} {2933773379069966367528193600},
b68=2933773379069966367528193600 is not divisible by 3, but all larger harmonic numbers have denominators divisible by 3.

You are given M(7) = 719102.

Find M(137).


和谐数分母的整除性

第n个和谐数Hn被定义为前n个正整数的倒数和,可以写成最简分数an/bn的形式。
$H_n = \displaystyle \sum_{k=1}^n \frac{1}{k} = \frac {a_n} {b_n}$, 其中$\text {gcd}(a_n, b_n)=1$。

令M(p)为使得bn不能被p整除的最大的n。

例如,M(3) = 68,因为H_{68} = \frac {a_{68}} {b_{68}} = \frac {14094018321907827923954201611} {2933773379069966367528193600},b68=2933773379069966367528193600不能被3整除,但是所有更大的和谐数的分母都能被3整除。

已知M(7) = 719102。

求M(137)。