Truncatable Prime
A zerofree number 
is called right
truncatable if 
and all numbers obtained by successively
removing the rightmost digits are prime.
There are exactly 83 right truncatable primes in base 10. The first few are 2, 3,
5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379,
593, 599, ... (OEIS A024770), the largest being
the 8-digit number 
(Angell and Godwin 1977). The
numbers of 
-digit right prime strings for 
, 2, ..., 8 are
4, 9, 14, 16, 15, 12, 8, and 5 (OEIS A050986;
Rivera puzzle 70).
Similarly, call a number 
left truncatable
if 
and all numbers obtained by successively
removing the leftmost digit are prime.
There are exactly 4260 left truncatable primes in base 10 when the digit zero is
not allowed. The first few are 2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83,
97, 113, 137, 167, 173, ... (OEIS A024785),
with the largest being the 24-digit number 
(Angell and Godwin 1977). The numbers of 
-digit left truncatable
primes for 
, 2, ... 24 are 4, 11, 39, 99, 192,
326, 429, 521, 545, 517, 448, 354, 276, 212, 117, 72, 42, 24, 13, 6, 5, 4, 3, and
1 (OEIS A050987; Rivera puzzle 70).
If zeros are permitted, the sequence of left truncatable primes is infinite, and the first few terms are 2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 137, 167, 173, 197, 223, 283, 307, ... (OEIS A033664).
J. Shallit has shown that in base 10, there is a finite, minimal list of primes that do not have any other primes as substrings (where digits do not need to be consecutive). This result is a special case of a much more general theorem, whose proof is unfortunately nonconstructive.
Call an 
-digit prime 
(with 
) is a restricted
left truncatable prime if
1. If the leftmost digit of 
is deleted,
a prime number 
is obtained for 
, and
2. No prime with 
digits can have its leftmost digit
removed to produce 
.
Kahan and Weintraub (1998) dub such primes "Henry VIII primes." Restricted left truncatable primes 
are therefore
a subset of left truncatable primes for which there are no left truncatable primes
of length 
having the same 
last digits as
. There are a total of 1440 such primes,
and the first few are 773, 3373, 3947, 4643, 5113, 6397, 6967, 7937, ... (OEIS A055521), the largest being 357686312646216567629137
(Angell and Godwin 1977, Kahan and Weintraub 1998).
Truncatable primes are also called Russian doll primes.
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