## Wednesday, May 17

### Countdown primes

The concatenation of the integers from 1 to n have been called Smarandache numbers, whereby the concatenation of the integers from n to 1 would be reverse Smarandache numbers. No Smarandache numbers are yet known to be prime but we have two for the reverse. I prefer to call them countdown primes.

The first is 82818079787776757473727170696867666564636261605958575655545352515049484746454443424140393837363534333231302928272625242322212019181716151413121110987654321, first noted by Ralf Stephan in 1998. The second countdown prime was found by Eric Weisstein in 2010. We can call them countdown(82) and countdown(37765) for short.

Surprisingly, a tabulation of countdown primes in bases other than ten appears not to have been tackled by anyone so I shall remedy that herewith:

2 — 2, 3, 4, 7, 11, 13, 25, 97, 110, 1939, ...
3 — 2, 5, 13, 57, 109, 638, 3069, ...
4 — 4, 106, 118, 130, 1690, ...
5 — 2, 313, 505, ...
6 — 2, 6, 17, 28, 33, 37, 81, 5611, ...
7 — 373, 1825, ...
8 — 2, 9, 47, 50, 99, 1969, 3672, ...
9 — 2, 5, 346, ...
10 — 82, 37765, ...
11 — 2, ...
12 — 3, 4, 5, 7, 17, 58, 106, 303, ...
13 — ?