On the Numbers of Ramanujan

¹ Akdeniz University, Department of Mathematics, Antalya 07058, Türkiye
² Institute of Mathematics, Henan Academy of Sciences, Zhengzhou 450046, China
³ Department of Mathematics, Institute of Science Tokyo, Meguro-ku, Tokyo 152-8551 Japan

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 9 December 2025
Accepted: 16 February 2026

Abstract

In Chapter 3 of his second notebook, Ramanujan defined numbers a (n, k) such that a (2, 0) = 1 and for n ≥ 2, a (n + 1, k) = (n − 1) a (n, k − 1) + (2n − 1 − k) a (n, k), where a(n, k) = 0 when k< 0 or k>n − 2. These numbers are expressed in terms of Stirling numbers of the first kind and associated Stirling numbers of the second kind, and satisfy certain divisibility properties. In this paper, we obtain further properties for a(n, k), including a new characterization of prime numbers. We also derive several congruences for the numbers of Ramanujan modulo p², some of which lead to new conditions for a prime number to be a Wilson prime.