A096246 Base-2 deletable primes (written in base 10).

2, 3, 5, 7, 11, 13, 19, 23, 29, 37, 43, 47, 53, 59, 61, 73, 79, 83, 101, 107, 109, 137, 149, 151, 157, 163, 167, 173, 179, 197, 211, 229, 277, 281, 293, 307, 311, 313, 317, 331, 347, 349, 359, 389, 397, 419, 421, 457, 461, 467, 557, 563, 569, 587, 599, 601, 613

Offset: 1

Author

Michael Kleber, Feb 28 2003

Keywords

• nonn
• base
• easy

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

• Lei Zhou, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A080608, A080603, A096235-A096245.

Programs

Maple

isDel := proc(n::integer) local b2,redu,rpr,d; if n = 2 or n =3 then RETURN(true); elif not isprime(n) then RETURN(false); else b2 := convert(n,base,2); for d from 1 to nops(b2) do redu := [op(1..d-1,b2),op(d+1..nops(b2),b2) ]; if op(nops(redu),redu) = 1 then rpr := sum( op(i,redu)*2^(i-1),i=1..nops(redu)); if isDel(rpr) then RETURN(true); fi; fi; od; RETURN(false); fi; end: for n from 1 to 200 do if isDel(ithprime(n)) then printf("%d,",ithprime(n)); fi; od: # R. J. Mathar, Apr 25 2006

Mathematica

a = {}; c = {1}; While[Length[a] < 100, b = c; c = {}; lb = Length[b]; Do[nb = b[[ib]]; cdb = RealDigits[nb, 2]; db = cdb[[1]]; ldb = cdb[[2]]; Do[dc = Insert[db, 0, j]; nc = FromDigits[dc, 2]; If[PrimeQ[nc], AppendTo[c, nc]], {j, 2, ldb + 1}]; Do[dc = Insert[db, 1, j]; nc = FromDigits[dc, 2]; If[PrimeQ[nc], AppendTo[c, nc]], {j, 2, ldb + 1}], {ib, 1, lb}]; c = Union[{}, c]; a = Union[a, c]]; a (* Lei Zhou, Mar 06 2015 *)
a = {0, 2}; d = {2, 3};
For[n = 3, n <= 15, n++,
  p = Select[Range[2^(n - 1), 2^n - 1], PrimeQ[#] &];
  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]]];  Break[]]]]];
d (* Robert Price, Nov 11 2018 *)

Python

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    if n == 2 or n == 3: return True
    b = bin(n)[2:]
    bi = (b[:i]+b[i+1:] for i in range(len(b)))
    return any(t[0] != '0' and ok(int(t, 2)) for t in bi)
print([k for k in range(614) if ok(k)]) # Michael S. Branicky, Jan 13 2022

Extensions

More terms from R. J. Mathar, Apr 25 2006