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.
