On some new arithmetic functions involving prime divisors and perfect powers

Integer division and perfect powers play a central role in numerous mathematical results, especially in number theory. Classical examples involve perfect squares like in Pythagora's theorem, or higher perfect powers as the conjectures of Fermat (solved in 1994 by A. Wiles [Wiles, A.J., Modular elliptic curves and Fermat's Last Theorem, Annals of Mathematics, 141 (1995), 443–551.]) or Catalan (solved in 2002 by P. Mihăilescu [Mihăilescu, P., Primary cyclotomic units and a proof of Catalan's conjecture, J. Reine Angew. Math., 572 (2004), 167–195.]). The purpose of this paper is two-fold. First, we present some new integer sequences $a\left(n\right)$, counting the positive integers smaller than n, having a maximal prime factor. We introduce an arithmetic function counting the number of perfect powers $i^j$ obtained for $1\le i,j\le n$. Along with some properties of this function, we present the sequence A303748, which was recently added to the Online Encyclopedia of Integer Sequences (OEIS) [The On-Line Encyclopedia of Integer Sequences, http://oeis.org, OEIS Foundation Inc. 2011.]. Finally, we discuss some other novel integer sequences.