## Friday, August 09, 2024

### Alphabetizing the integers

Last month I wrote an article called "Integers, in alphabetical order" that illustrated my then-obsession with the topic. In February 1981, Ross Eckler had an article called "Alphabetizing the integers" (in the magazine Word Ways) that loosely anticipated my exploration.

On page 20 of his article, Eckler notes Philip Cohen's approach to the "Bergerson" problem (find the fixed points for integers 1 to n):

one   one   one     four    five    five    five    eight   eight
two   three   one     four    four    four    five    five
two     three   one     one     one     four    four
two     three   six     seven   one     nine
two     three   six     seven   one
two     three   six     seven
two     three   six
two     three
two

The number of fixed-point counts leads us to OEIS sequence A340671, which provides the actual fixed points in one of the links. Eckler gives us the list-sizes which produce matches for the first twenty integers. I have extended this listing to 100000 integers here. Eckler then notes that 3 appears (I am paraphrasing) in A340671 at positions 13 and 31 and asks for additional values. The third occurrence is at position 201 (then 203, 204, 208, ...). He didn't ask for a first 4, 5, 6, ... appearance as these were clearly beyond his realm of realizability at the time. But here they are:

1        1
2        2
3       13
4      202
5      213
6     2202
7     2213
8    14475
9   233164
10   320200
11   449694
12  2450694
13  4367488
14  4580804
15  4580824

The listing is for the first appearance of 1 to 15, giving the position/index at which it occurs. Michael Branicky found #12 on July 7. I determined #13 to #15 yesterday. The fixed points are added here.