Harmonic prime

Last updated May 10, 2025

A harmonic prime(sequence A092101 in the OEIS ) is a prime number that divides the numerators of exactly three harmonic numbers.

Specifically, a harmonic prime $p$ is always a factor of the numerators of the partial harmonic sums at positions $p - 1$ , $p 2 - p$ , and $p 2 - 1$ .

For example, the numerators of the fractions given by $\sum _{i=1}^{4}{\frac {1}{i}}$ , $\sum _{i=1}^{20}{\frac {1}{i}}$ , and $\sum _{i=1}^{24}{\frac {1}{i}}$ are 25, 55835135, and 1347822955, each of which is divisible by 5.

All prime numbers greater than 5 can also be found at those three indices, but many also appear at other indices. It is conjectured that there are infinitely many harmonic primes. ^[1]

References

↑ Boyd, D. W. (1994). "A p-adic Study of the Partial Sums of the Harmonic Series". Experimental Mathematics . 3 (4): 287–302. doi:10.1080/10586458.1994.10504298. Zbl 0838.11015. CiteSeerX: 10.1.1.56.7026 . Archived from the original on 27 January 2016.

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[1] Boyd, D. W. (1994). "A p-adic Study of the Partial Sums of the Harmonic Series". Experimental Mathematics . 3 (4): 287–302. doi:10.1080/10586458.1994.10504298. Zbl 0838.11015. CiteSeerX: 10.1.1.56.7026 . Archived from the original on 27 January 2016.

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