A096236 -id:A096236 - cp's OEIS frontend

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, 98948426, 193629933, 379398057, 744508765, 1461801309

Offset: 1

Michael Kleber, Feb 28 2003

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.

			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

Cf. A080608, A080603, A096236-A096246.

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 *)

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

a(19)-a(30) from Ryan Propper, Jul 18 2005
a(31)-a(33) from Michael S. Branicky, Jan 14 2022
a(34)-a(37) from Michael S. Branicky, May 30 2025
a(38) from Michael S. Branicky, Jun 02 2025

A101216 Number of n-digit base-2 deletable digit-sum multiple (DSM) integers.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 13, 26, 44, 87, 165, 305, 523, 948, 1792, 3501, 6644, 12622, 23334, 43232, 79651, 149716, 281278, 532051, 1000247, 1883093, 3577619, 6901273, 13495425, 26522993, 51976835

Offset: 1

John W. Layman, Dec 14 2004

A positive integer n is a base-b digit-sum-multiple (DSM) number if the sum of the digits of n, in base b, divides n. It is a deletable base-b DSM if it has the property that removing some digit leaves either the empty string or another deletable base-b DSM.

Cf. A096236.

from itertools import count, islice
def ok(n, prevset):
    b = bin(n)[2:]
    if n%b.count("1"): return False
    si = (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 si)
def agen(): # generator of terms
    s, snxt = {1}, set()
    for n in count(2):
        yield len(s)
        for i in range(2**(n-1), 2**n):
            if ok(i, s):
                snxt.add(i)
        s, snxt = snxt, set()
print(list(islice(agen(), 20))) # Michael S. Branicky, Feb 25 2023

a(19)-a(32) from Michael S. Branicky, Feb 25 2023

cp's OEIS Frontend

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

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