On an iterated arithmetic function problem of Erdos and Graham

Abstract

Erdős and Graham define $g(n)=n+\ddot{I}•(n)$ and the iterated application $g_k(n)=g(g_{k-1}(n))$. They ask for solutions of $g_{k+r}(n)=2g_k(n)$ and observe $g_{k+2}(10)=2g_k(10)$ and $g_{k+2}(94)=2g_k(94)$. We show that understanding the case $r=2$ is equivalent to understanding all solutions of the equation $\ddot{I}•(n)+\ddot{I}•(n+\ddot{I}•(n))=n$ and find the explicit solutions $n=2^{\ell }\cdot \left\{1,3,5,7,35,47\right\}$. This list of solutions is possibly complete: any other solution derives from a number $n=2^{\ell }p$ where $p\ge 10^{10}$ is a prime satisfying $\ddot{I}•((3p-1)/4)=(p+1)/2$. Primes with this property seem to be very rare and maybe no such prime exists.