A080608 Deletable primes: primes such that removing some digit leaves either the empty string or another deletable prime (possibly preceded by some zeros).

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

David W. Wilson, Feb 25 2003

Keywords

• nonn
• easy
• base

Comments

Subsequence of A179336. - Reinhard Zumkeller, Jul 11 2010

Leading zeros are allowed in the number that appears after the digit is deleted. For example the prime 100003 is deletable because of the sequence 00003, 0003, 003, 03, 3 consists of primes. Because of this, it appears that deletable primes are relatively common in the region just above a power of ten. For example 10^1000 + 2713 is a deletable prime. - Jeppe Stig Nielsen, Aug 01 2018

For a version that does not allow leading zeros, see A305352. - Jeppe Stig Nielsen, Aug 01 2018

Examples

			410256793 is a deletable prime since each member of the sequence 410256793, 41256793, 4125673, 415673, 45673, 4567, 467, 67, 7 is prime (Weisstein, Caldwell).
		

Links

• David W. Wilson, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Deletable Prime

Crossrefs

Cf. A080603, A096235-A096246, A305352.

Programs

Maple

read("transforms"):
isA080608 := proc(n)
    option remember;
    local dgs,i ;
    if isprime(n) then
        if n < 10 then
            true;
        else
            dgs := convert(n,base,10) ;
            for i from 1 to nops(dgs) do
                subsop(i=NULL,dgs) ;
                digcatL(ListTools[Reverse](%)) ;
                if procname(%)  then
                    return true;
                end if;
            end do:
            false ;
        end if;
    else
        false;
    end if;
end proc:
n := 1;
for i from 1 to 500 do
    p := ithprime(i) ;
    if isA080608(p) then
        printf("%d %d\n",n,p) ;
        n := n+1 ;
    fi ;
end do: # R. J. Mathar, Oct 11 2014

Mathematica

Rest@ Union@ Nest[Function[{a, p}, Append[a, With[{w = IntegerDigits[p]}, If[# == True, p, 0] &@ AnyTrue[Array[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,is(fromdigits(vecextract(d,Str("^"i))))&&return(1));0 \\ Jeppe Stig Nielsen, Aug 01 2018

Perl

use ntheory ":all"; sub is { my $n=shift; return 0 unless is_prime($n); my @d=todigits($n); return 1 if @d==1; is(fromdigits([vecextract(\@d,~(1<<$))])) && return 1 for 0..$#d; 0; } # _Dana Jacobsen, Nov 16 2018

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(ok(int(t)) for t in si)
print([k for k in range(440) if ok(k)]) # Michael S. Branicky, Jan 28 2023