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A096235 Number of n-bit base-2 deletable primes. 34
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 (list; graph; refs; listen; history; text; internal format)
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: A038715 A293518 A057040 * A147851 A321380 A218694

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

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Last modified June 30 12:59 EDT 2022. Contains 354939 sequences. (Running on oeis4.)