A096235 Number of n-bit base-2 deletable primes.

0, 2, 2, 2, 3, 6, 6, 11, 18, 31, 49, 87, 155, 253, 427, 781, 1473, 2703, 5094, 9592, 18376, 35100, 67183, 129119, 249489, 482224, 930633, 1803598, 3502353, 6813094, 13271996, 25892906, 50583039

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. However, in base 2 we adopt the convention that 2 = 10 and 3 = 11 are deletable.

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..33.

Example

d base-2 d-digit deletable primes
2 2=10, 3=11
3 5=101, 7=111
4 11=1011, 13=1101
5 19=10011, 23=10111, 29=11101
6 37=100101, 43=101011, 47=101111, 53=110101, 59=111011, 61=111101
7 73=1001001, 79=1001111, 83=1010011, 101=1100101, 107=1101011, 109=1101101

Mathematica

a = {0, 2}; d = {2, 3};
For[n = 3, n <= 15, n++,
p = Select[Range[2^(n - 1), 2^n - 1], PrimeQ[#] &];
ct = 0;
For[i = 1, i <= Length[p], i++,
  c = IntegerDigits[p[[i]], 2];
  For[j = 1, j <= n, j++,
   t = Delete[c, j];
   If[t[[1]] == 0, Continue[]];
   If[MemberQ[d, FromDigits[t, 2]], AppendTo[d, p[[i]]]; ct++;
     Break[]]]];
AppendTo[a, ct]];
a (* Robert Price, Nov 11 2018 *)

Prog

(Python)
from sympy import isprime
def ok(n, prevset):
    if not isprime(n): return False
    b = bin(n)[2:]
    bi = (b[:i]+b[i+1:] for i in range(len(b)))
    return any(t[0] != '0' and int(t, 2) in prevset for t in bi)
def afind(terms):
    s, snxt = {2, 3}, set()
    print("0, ", len(s), end=", ")
    for n in range(3, terms+1):
        for i in range(2**(n-1), 2**n):
            if ok(i, s):
                snxt.add(i)
        s, snxt = snxt, set()
        print(len(s), end=", ")
afind(20) # Michael S. Branicky, Jan 14 2022

Crossrefs

Cf. A080608, A080603, A096236-A096246.

Sequence in context: A293518 A057040 A363772 * A147851 A367088 A321380

Adjacent sequences: A096232 A096233 A096234 * A096236 A096237 A096238

Keyword

nonn,base,more

Author

Michael Kleber, Feb 28 2003

Extensions

a(19)-a(30) from Ryan Propper, Jul 18 2005

a(31)-a(33) from Michael S. Branicky, Jan 14 2022

Status

approved