A096236 Number of n-digit base-3 deletable primes.

1, 2, 4, 7, 13, 24, 38, 72, 122, 226, 400, 684, 1246, 2381, 4384, 8330, 15839, 30617, 58764, 113987, 221994, 434498, 852036, 1673320, 3296641, 6509179, 12880606, 25513097, 50592761, 100474328, 199784415, 397803026, 792771729, 1581115299, 3154049796

Offset

1,2

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.

Links

Table of n, a(n) for n=1..35.

Daniel Okwor, Deletable primes in bases 3 through 12: code and verification, 2026.

Mathematica

b = 3; a = {1}; d = {2};
For[n = 2, n <= 10, n++,
  p = Select[Range[b^(n - 1), b^n - 1], PrimeQ[#] &];
  ct = 0;
  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]]]; ct++;
     Break[]]]];
  AppendTo[a, ct]];
a (* Robert Price, Nov 12 2018 *)

Prog

(Python)
from sympy import isprime
from sympy.ntheory.digits import digits
def ok(n, prevset, base=3):
    if not isprime(n): return False
    s = "".join(str(d) for d in digits(n, base)[1:])
    si = (s[:i]+s[i+1:] for i in range(len(s)))
    return any(t[0] != '0' and int(t, base) in prevset for t in si)
def afind(terms):
    s, snxt, base = {2}, set(), 3
    print(len(s), end=", ")
    for n in range(2, terms+1):
        for i in range(base**(n-1), base**n):
            if ok(i, s):
                snxt.add(i)
        s, snxt = snxt, set()
        print(len(s), end=", ")
afind(13) # Michael S. Branicky, Jan 14 2022

Crossrefs

Cf. A080608, A080603, A096235-A096246.

Sequence in context: A005595 A296689 A327543 * A356932 A002574 A069765

Adjacent sequences: A096233 A096234 A096235 * A096237 A096238 A096239

Keyword

nonn,more,base,changed

Author

Michael Kleber, Feb 28 2003

Extensions

a(6)-a(15) from John W. Layman, Dec 14 2004

a(16)-a(26) from Ryan Propper, Jul 19 2005

a(27)-a(35) from Daniel Okwor, Jun 12 2026

Status

approved