# Besides our we sometimes use

There is something special about these three numbers. Many other numbers fit this list but these three should be the most common ones.

• $$2$$

• $$1666$$

• $$18364758544493064720$$

What is the hidden rule behind the creation of these numbers?

• The help center for this site is still sitting on the boilerplate after 6 years, so I'm not sure whether the default rules allow for using OEIS; could you let me know? Oct 25 '20 at 6:36
• Sure, you can use it. But it probably won't be helpful Oct 25 '20 at 8:00
• It got me as far as the posted answer. ;-) Oct 25 '20 at 17:26
• @chrylis-cautiouslyoptimistic my answer did not come from OEIS, but originally from pencil, paper and calculator. The expanded answer came from a short C program. When people post number sequence questions that can be solved by consulting OEIS there is often rather more negative commenting. Oct 26 '20 at 17:04

What is the hidden rule behind the creation of these numbers?

Rule: all digits descending in common number systems, as written in Base 10.

Credit WeatherVane for finding the link to number systems and 2 out of 3. We can close out this puzzle by solving 1666 and thereby confirming the hidden rule:

System one: Base 2: 102 = 210
System two: Roman Numerals: MDCLXVI = 166610
System three: Base 16: FEDCBA987654321016 = 1836475854449306472010

This explains the title as well:

Besides our NUMBER SYSTEM we sometimes use OTHER NUMBER SYSTEMS.

I can't get the first two, but for the third

1836475854449306472010 = FFFFFFFFFFFFFFFF16 - 0123456789ABCDEF16

I guess there is a similar relationship for other number bases.

Edit:

Simplified to use just descending digits in each number base

2 102 = 210
3 2103 = 2110
4 32104 = 22810
5 432105 = 293010
6 5432106 = 4479010
7 65432107 = 80066710
8 765432108 = 1643482410
9 8765432109 = 38136704410
10 987654321010 = 987654321010
11 A987654321011 = 28245855390510
12 BA987654321012 = 884241366769210
13 CBA987654321013 = 30077180724091810
14 DCBA987654321014 = 1104625530588015810
15 EDCBA987654321015 = 43565973787891621510
16 FEDCBA987654321016 = 1836475854449306472010

But I still have not solved the second part.

• on the right track, but rot13(lbh jrag shegure guna vagraqrq. Jung qb lbh frr vs lbh bayl sbphf ba gur pbairefvba?) Oct 25 '20 at 9:14
• Yes, the second part is the trickiest Oct 25 '20 at 9:45
• What does the question title mean? Oct 25 '20 at 9:51
• Besides our rot13(ahzore flfgrz).... Oct 25 '20 at 10:03