The p-adic Valuation of Eulerian Numbers: Trees and Bernoulli Numbers

Abstract

In this work, the p-adic valuation of Eulerian numbers is explored. A tree whose nodes contain information about the p-adic valuation of these numbers is constructed, and this tree, along with some classical results for Bernoulli numbers, is used to compute the exact p divisibility for the Eulerian numbers when the first variable lies in a congruence class and p satisfies some regularity properties.

2000 AMS Subject Classification::

• Primary 11B50
• Secondary 11B75
• 11B68

Keywords:

• p-adic valuation
• Eulerian numbers
• Bernoulli numbers
• Wolstenholme primes
• regular primes

Notes

¹Available at http://oeis.org/.

²The Mathematica implementation can be found at http://emmy.uprrp.edu/lmedina/software/.