Journal of Integer Sequences, Vol. 15 (2012), Article 12.3.1

A Study of a Curious Arithmetic Function


Bakir Farhi
Department of Mathematics
University of Béjaia
Béjaia
Algeria

Abstract:

In this note, we study the arithmetic function $ f : \mathbb{Z}_+^* \rightarrow \mathbb{Q}_+^*$ defined by $ f(2^k \ell) = \ell^{1 - k}$ ( $ \forall k , \ell \in \mathbb{N}$, $ \ell$ odd). We show several important properties about this function, and we use them to obtain some curious results involving the $ 2$-adic valuation. In the last section of the paper, we generalize those results to any other $ p$-adic valuation.

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(Concerned with sequences A185021 A185275.)

Received April 27 2011; revised version received October 15 2011; January 25 2012. Published in Journal of Integer Sequences, January 28 2012.

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