Closest Harmonic Number to an Integer
21 days ago
- #Python
- #Harmonic Numbers
- #Numerical Approximation
- Harmonic numbers Hn are never integers for n > 1.
- Two problems to solve: accurately compute harmonic numbers and find n such that Hn is closest to a given integer m.
- For small n, compute Hn directly; for large n, use an asymptotic approximation involving the Euler-Mascheroni constant γ.
- To find n for Hn closest to m, approximate n as exp(m − γ).
- Python code provided to compute Hn and find the nearest harmonic number to a given m.
- Example usage: nearest_harmonic_number(10) returns 12366 with H(12366) ≈ 9.99996.
- Code also works with real numbers, e.g., finding Hn closest to √20.