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

1,1

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

Links

David W. Wilson, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Deletable Prime

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A080603, A096235-A096246, A305352.

Sequence in context: A247980 A234851 A179336 * A305352 A347864 A137812

Adjacent sequences: A080605 A080606 A080607 * A080609 A080610 A080611

Keyword

nonn,easy,base

Author

David W. Wilson, Feb 25 2003

Status

approved