A125590 Largest n-digit base-10 deletable prime.

7, 97, 997, 9973, 99929, 999907, 9999907, 99999307, 999996671, 9999996073, 99999966307, 999999908773, 9999999710639, 99999999697769, 999999997160639, 9999999996977699, 99999999980803477, 999999999961861807, 9999999999961861807, 99999999999807429133

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

Table of n, a(n) for n=1..20.

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]

Index entries for sequences related to truncatable primes

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

Cf. A080608, A096243, A096246, A125589.

Sequence in context: A178007 A241206 A127892 * A068694 A158579 A003618

Adjacent sequences: A125587 A125588 A125589 * A125591 A125592 A125593

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