OFFSET
1,1
COMMENTS
A prime p is a base-b deletable prime if when written in base b it has the property that removing some digit leaves either the empty string or another deletable prime. "Digit" means digit in base b.
Deleting a digit cannot leave any leading zeros in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.
REFERENCES
C. Caldwell, Truncatable primes, J. Recreational Math., 19:1 (1987) 30-33. [Discusses left truncatable primes, right truncatable primes and deletable primes.]
LINKS
I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
C. Caldwell, Deletable primes
Prime Curios, A 300-digit example
Carlos Rivera, Puzzle 138: Deletable Primes, Prime Puzzles and Problems Connection. [Includes a 500-digit example]
EXAMPLE
99929 -> 9929 -> 929 -> 29 -> 2.
MATHEMATICA
b = 10; a = {7}; d = {2, 3, 5, 7};
For[n = 2, n <= 5, n++,
p = Select[Range[b^(n - 1), b^n - 1], PrimeQ[#] &];
For[i = 1, i <= Length[p], i++,
c = IntegerDigits[p[[i]], b];
For[j = 1, j <= n, j++,
t = Delete[c, j];
If[t[[1]] == 0, Continue[]];
If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]]; Break[]]]];
AppendTo[a, Last[d]]];
a (* Robert Price, Nov 13 2018 *)
PROG
(Python)
from sympy import isprime, prevprime
from functools import cache
@cache
def deletable_prime(n):
if not isprime(n): return False
if n < 10: return True
s = str(n)
si = (s[:i]+s[i+1:] for i in range(len(s)))
return any(t[0] != '0' and deletable_prime(int(t)) for t in si)
def a(n):
p = prevprime(10**n)
while not deletable_prime(p): p = prevprime(p)
return p
print([a(n) for n in range(1, 15)]) # Michael S. Branicky, Jan 13 2022
CROSSREFS
KEYWORD
nonn,base
AUTHOR
N. J. A. Sloane, Jan 07 2007
EXTENSIONS
a(6)-a(8) from Michael Kleber, Jan 08 2007
a(9)-a(16) from Joshua Zucker, May 11 2007
a(17)-a(20) from Michael S. Branicky, Jan 13 2022
STATUS
approved