A319596 Base-3 deletable primes (written in base 10).

2, 5, 7, 11, 17, 19, 23, 29, 47, 53, 59, 61, 71, 73, 83, 89, 101, 107, 137, 167, 173, 179, 181, 191, 197, 223, 233, 251, 263, 269, 317, 431, 461, 491, 503, 509, 521, 541, 547, 557, 569, 587, 593, 653, 659, 673, 677, 683, 701, 709, 719, 809, 911, 947, 953

Offset: 1

Author

Robert Price, Nov 14 2018

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.

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

• Robert Israel, Table of n, a(n) for n = 1..10000 (first 177 terms from Robert Price)

Crossrefs

Cf. A080608, A080603, A096235-A096246.

Programs

Maple

S:= {2}: count:= 0:
p:= 2;
while count < 200 do
  p:= nextprime(p);
  d:= floor(log[3](p));
  for i from 0 to d do
    x:= p mod 3^(i+1);
    q:= (x mod 3^i) + (p-x)/3;
    if q >= 3^(d-1) and member(q,S) then
       S:= S union {p}; count:= count+1; break
      fi
  od;
od:
sort(convert(S,list)); # Robert Israel, Nov 26 2020

Mathematica

b = 3; d = {};
p = Select[Range[2, 10000], PrimeQ[#] &];
For[i = 1, i <= Length[p], i++,
  c = IntegerDigits[p[[i]], b];
  If[Length[c] == 1, AppendTo[d, p[[i]]]; Continue[]];
  For[j = 1, j <= Length[c], j++,
   t = Delete[c, j];
   If[t[[1]] == 0, Continue[]];
   If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]]; Break[]]]];
d (* Robert Price, Dec 05 2018 *)

Python

from sympy import isprime
from sympy.ntheory.digits import digits
def ok(n, base=3):
    if not isprime(n): return False
    if n < 3: return True
    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 ok(int(t, base)) for t in si)
print([k for k in range(954) if ok(k)]) # Michael S. Branicky, Jan 14 2022

Extensions

Removed the term 3. As pointed out by Kevin Ryde, there is no need to "seed" the list using base-2 assumptions. - Robert Price, Dec 05 2018