https://www.johndcook.com/blog/2025/11/19/closest-harmonic-number-to-an-integer/

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Closest harmonic number to an integer

Posted on 19 November 2025 by John

I mentioned in the previous post that the harmonic numbers H[n] are
never integers for n > 1. In the spirit of that post, I'd like to
find the value of n such that H[n] is closest to a given integer m.

We have two problems to solve. First, how do we accurately and
efficiently compute harmonic numbers? For small n we can directly
implement the definition. For large n, the direct approach would be
slow and would accumulate floating point error. But in that case we
could use the asymptotic approximation

H_n \approx \log n + \gamma + \frac{1}{2n} - \frac{1}{12n^2}

from this post. As is often the case, the direct approach gets worse
as n increases, but the asymptotic approximation gets better as n
increases. Here g is the Euler-Mascheroni constant.

The second problem to solve is how to find the value of n so that H[n
] comes closest to m without trying too many possible values of n? We
can discard the higher order terms above and see that n is roughly
exp(m - g).

Here's the code.

import numpy as np

gamma = 0.57721566490153286

def H(n):
    if n < 1000:
        return sum([1/k for k in range(1, n+1)])
    else:
        n = float(n)
        return np.log(n) + gamma + 1/(2*n) - 1/(12*n**3)

# return n such that H_n is closest harmonic number to m
def nearest_harmonic_number(m):

    if m == 1:
        return 1

    guess = int(np.exp(m - gamma))
    if H(guess) < m:
        i = guess
        while H(guess) < m: guess += 1 j = guess else: j = guess while H(guess) > m:
            guess -= 1
        i = guess

    x = np.array([abs(H(k) - m) for k in range(i, j+1)])
    return i + np.argmin(x)

We can use this, for example, to find the closest harmonic number to
10.

>>> nearest_harmonic_number(10)
12366
>>> H(12366)
9.99996214846655

I wrote the code with integer values of m in mind, but the code works
fine with real numbers. For example, we could find the harmonic
number closest to [?]20.

>>> nearest_harmonic_number(20**0.5)
49
>>> H(49)**2
20.063280462918804

Related posts

  * Harmonic e
  * A strange take on the harmonic series
  * Computing g

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John D. Cook

John D. Cook, PhD

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