Journal of Integer Sequences, Vol. 18 (2015), Article 15.9.7 |

Massachusetts Institute of Technology

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**Abstract:**

This paper deals with those positive integers *N* such that, for
given integers *g* and *k* with 2 ≤ *k* < *g*, the
base-*g* digits of *kN* appear in reverse order from those of
*N*. Such *N* are called (*g*, *k*) reverse
multiples. Young, in 1992, developed a kind of tree reflecting
properties of these numbers; Sloane, in 2013, modified these trees into
directed graphs and introduced certain combinatorial methods to
determine from these graphs the number of reverse multiples for given
values of *g* and *k* with a given number of digits. We prove
Sloanes isomorphism conjectures for 1089 graphs and complete graphs,
namely that the Young graph for *g* and *k* is a 1089 graph
if and only if *k*+1 | *g* and is a complete Young graph on
*m* nodes if and only if ⌊ gcd(*g* - *k*, *k*^{2} - 1)/(*k* + 1) ⌋ = *m* - 1. We also extend his study of cyclic
Young graphs and prove a minor result on isomorphism and the nodes
adjacent to the node [0, 0].

(Concerned with sequences A001232 A008918 A008919 A031877 A169824.)

Received April 10 2015;
revised version received August 16 2015; August 20 2015.
Published in *Journal of Integer Sequences*, August 21 2015.

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