A305352 Deletable primes (A080608) under the stricter rule that leading zeros are disallowed.

2, 3, 5, 7, 13, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 103, 107, 113, 127, 131, 137, 139, 157, 163, 167, 173, 179, 193, 197, 223, 229, 233, 239, 263, 269, 271, 283, 293, 307, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 397, 431, 433, 439

Offset: 1

Author

Jeppe Stig Nielsen, Aug 01 2018

Keywords

• nonn
• base

Comments

Subset of A080608. Numbers 2003, 2017, 2053, 3023, ... are in A080608 but not here.

If you start from a one-digit prime, you can try to build larger and larger deletable primes by inserting digits. If at one point you get stuck and cannot enlarge the number, you have reached a non-insertable prime (A125001). - Jeppe Stig Nielsen, Mar 28 2021

Examples

			2003 is not a member since removing a digit will either give 003 which has a leading zero, or give one of the numbers 203 or 200 which are both composite. However, 2003 is in A080608 because all of 2003, 003, 03, 3 are prime.
		

Links

• Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A080608, A080603, A096235-A096246, A125001, A322474.

Programs

Mathematica

Rest@ Union@ Nest[Function[{a, p}, Append[a, With[{w = IntegerDigits[p]}, If[# == True, p, 0] &@ AnyTrue[Array[If[First@ # == 0, 1, FromDigits@ #] &@ Delete[w, #] &, Length@ w], ! FreeQ[a, #] &]]]] @@ {#, Prime[Length@ # + 1]} &, Prime@ Range@ PrimePi@ 10, 81] (* Michael De Vlieger, Aug 02 2018 *)

PARI

is(n) = !ispseudoprime(n)&&return(0);my(d=digits(n));#d==1&&return(1);for(i=1,#d,my(v=vecextract(d,Str("^"i)));v[1]!=0&&is(fromdigits(v))&&return(1));0

Python

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    if n < 10: return True
    s = str(n)
    si = (s[:i]+s[i+1:] for i in range(len(s)))
    return any(t[0] != '0' and ok(int(t)) for t in si)
print([k for k in range(440) if ok(k)]) # Michael S. Branicky, Jan 28 2023