[seqfan] Leyland Numbers Puzzle

[Hans Havermann]

https://oeis.org/A076980

I’ve taken on a somewhat challenging task to index (by size) all the Leyland numbers less than 10^100000 (there are 331682621) so that I might assign (most of) the known Leyland primes with a Leyland-number index. The vast majority of the numbers are of course too large to represent in decimal but that’s ok; we can represent them as (x,y) pairs instead:

1 (2,1)
2 (2,2)
3 (3,2)
4 (4,2)
5 (3,3)
6 (5,2)
7 (6,2)
8 (4,3)
9 (7,2)
10 (8,2)
etc.

It’s easy enough to generate the pairs but how do I sort them? I decided to attach to each pair an approximation of the Leyland number that the pair represents and then sort that. Doing so, it turns out that there are millions of still-unsorted trouble spots where the approximations of anywhere from 2 to 13 consecutive "sorted" numbers are equal. I’m in the process of fixing that but here’s an interesting thing that I noticed. The 13 approximately equal {(240240,2), (120120,4), (80080,8), (60060,16), (48048,32), (40040,64), (34320,128), (30030,256), (24024,1024), (21840,2048), (20020,4096), (18480,8192), (17160,16384)} Leyland pairs are conspicuously missing (x,512). Can anyone put their finger on why that should be?