examples of 1-automorphic numbers

Concerning ourselves only with searching for automorphic numbers $n$ in bases $1<b<17$ (binary to hexadecimal) and the ranges given by the iterator $0<i<b^{3}-1$ , and limiting to $m=1$ we find the following results:

First, it is obvious that 1 is a 1-automorphic number regardless of the base.

For the range and limit specified, there are no other 1-automorphic numbers in binary through quinary, bases 7 through 9, 11, 13 and 16.

In base 6, there are 1, 3, 4, 9, 28, 81, 136, and it is easy to verify that ${3_{6}}^{2}=13_{6}$ , ${4_{6}}^{2}=24_{6}$ , ${13_{6}}^{2}=213_{6}$ , ${44_{6}}^{2}=3344_{6}$ , etc.

In base 10, these ought to look familiar: 1, 5, 6, 25, 76, 376, 625.

Duodecimal: 1, 4, 9, 64, 81, 513, 1216. Noticing that 4 also appears in the list for base 6, we might wonder if 4 is always 1-automorphic when $6|b$ ? The question is moot because the next multiple of 6 is $18>4^{2}$ , thus in base 18 and any other higher bases, the square of 4 is also a 1-digit number.

Base 14: 1, 7, 8, 49, 148, 344, 2401.

Base 15: 1, 6, 10, 100, 126, 1000, 2376. Base 15 is the smallest odd base $b$ to have 1-automorphic numbers in the range specified. This should not be taken to imply that it is the smallest odd base to have automorphic numbers at all.

Title	examples of 1-automorphic numbers
Canonical name	ExamplesOf1automorphicNumbers
Date of creation	2013-03-22 16:20:20
Last modified on	2013-03-22 16:20:20
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Example
Classification	msc 11A63