John Cook: Numerators of harmonic numbers

Harmonic numbers

The nth harmonic number, H_n, is the sum of the reciprocals of the integers up to and including n. For example,

H₄ = 1 + 1/2 + 1/3 + 1/4 = 25/12.

Here’s a curious fact about harmonic numbers, known as Wolstenholme’s theorem:

For a prime p> 3, the numerator of H_p-1 is divisible by p².

The example above shows this for p = 5. In that case, the numerator is not just divisible by p², it isp², though this doesn’t hold in general. For example, H₁₀ = 7381/2520. The numerator 7381 is divisible by 11² = 121, but it’s not equal to 121.

Generalized harmonic numbers

The generalized harmonic numbers H_n,m are the sums of the reciprocals of the first n positive integers, each raised to the power m. Wolstenholme’s theorem also says something about these numbers too:

For a prime p> 3, the numerator of H_p-1,2 is divisible by p.

For example, H_4,2 = 205/144, and the numerator is clearly divisible by 5.

Computing with Python

You can play with harmonic numbers and generalized harmonic numbers in Python using SymPy. Start with the import statement

from sympy.functions.combinatorial.numbers import harmonic

Then you can get the nth harmonic number with harmonic(n) and generalized harmonic numbers with harmonic(n, m).

To extract the numerators, you can use the method as_numer_denom to turn the fractions into (numerator, denominator) pairs. For example, you can create a list of the denominators of the first 10 harmonic numbers with

[harmonic(n).as_numer_denom()[0] for n in range(10)]

What about 0?

You might notice that harmonic(0) returns 0, as it should. The sum defining the harmonic numbers is empty in this case, and empty sums are defined to be zero.