On a Family of Sequences Related to Prime Counting Function

Abstract

In this study, we introduce a new family of integer sequences which are related to prime-counting function and we focus on some properties of these sequences. Sequence A316434 in OEIS is the fundamental member of solution family that we study. More precisely, we investigate the solutions of recurrence a(n) = a(π(n)) + a(n-π(n)) with some natural initial conditions where π(n) is defined by A000720 in OEIS.