A125590 - OEIS

%I #30 Apr 03 2023 10:36:11

%S 7,97,997,9973,99929,999907,9999907,99999307,999996671,9999996073,

%T 99999966307,999999908773,9999999710639,99999999697769,

%U 999999997160639,9999999996977699,99999999980803477,999999999961861807,9999999999961861807,99999999999807429133

%N Largest n-digit base-10 deletable prime.

%C 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.

%C 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.

%D C. Caldwell, Truncatable primes, J. Recreational Math., 19:1 (1987) 30-33. [Discusses left truncatable primes, right truncatable primes and deletable primes.]

%H I. O. Angell and H. J. Godwin, <a href="http://dx.doi.org/10.1090/S0025-5718-1977-0427213-2">On Truncatable Primes</a>, Math. Comput. 31, 265-267, 1977.

%H C. Caldwell, <a href="https://t5k.org/glossary/page.php?sort=DeletablePrime">Deletable primes</a>

%H Prime Curios, <a href="https://t5k.org/curios/page.php?curio_id=1420">A 300-digit example</a>

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_138.htm">Puzzle 138: Deletable Primes</a>, Prime Puzzles and Problems Connection. [Includes a 500-digit example]

%H <a href="/index/Tri#tprime">Index entries for sequences related to truncatable primes</a>

%e 99929 -> 9929 -> 929 -> 29 -> 2.

%t b = 10; a = {7}; d = {2, 3, 5, 7};

%t For[n = 2, n <= 5, n++,

%t   p = Select[Range[b^(n - 1), b^n - 1], PrimeQ[#] &];

%t   For[i = 1, i <= Length[p], i++,

%t    c = IntegerDigits[p[[i]], b];

%t    For[j = 1, j <= n, j++,

%t     t = Delete[c, j];

%t     If[t[[1]] == 0, Continue[]];

%t     If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]]; Break[]]]];

%t   AppendTo[a, Last[d]]];

%t a (* _Robert Price_, Nov 13 2018 *)

%o (Python)

%o from sympy import isprime, prevprime

%o from functools import cache

%o @cache

%o def deletable_prime(n):

%o     if not isprime(n): return False

%o     if n < 10: return True

%o     s = str(n)

%o     si = (s[:i]+s[i+1:] for i in range(len(s)))

%o     return any(t[0] != '0' and deletable_prime(int(t)) for t in si)

%o def a(n):

%o     p = prevprime(10**n)

%o     while not deletable_prime(p): p = prevprime(p)

%o     return p

%o print([a(n) for n in range(1, 15)]) # _Michael S. Branicky_, Jan 13 2022

%Y Cf. A080608, A096243, A096246, A125589.

%K nonn,base

%O 1,1

%A _N. J. A. Sloane_, Jan 07 2007

%E a(6)-a(8) from _Michael Kleber_, Jan 08 2007

%E a(9)-a(16) from _Joshua Zucker_, May 11 2007

%E a(17)-a(20) from _Michael S. Branicky_, Jan 13 2022