A193890 Primes p such that replacing any single decimal digit d with 3*d produces another prime (obviously p can contain only digits 0, 1, 2 or 3).

11, 311, 1301, 10133, 1030031

Offset: 1

Author

Arkadiusz Wesolowski, Aug 08 2011

Keywords

• nonn
• base
• hard
• more

Comments

These numbers do not occur in A050249 (weakly associated primes).

p cannot contain digits 1 and 2 at the same time due to divisibility by 3.

No more terms < 10^9. [Reinhard Zumkeller, Aug 11 2011]

No more terms < 10^14. - Arkadiusz Wesolowski, Feb 08 2012

No more terms < 10^18. - Giovanni Resta, Feb 23 2013

No more terms < 10^22. - Jan van Delden, Mar 06 2016

The number of occurrences of the digit 1 or 2 is congruent to 2 (mod 3). - Robert Israel, Mar 07 2016

Examples

			1301 belongs to this sequence because 1303, 1301, 1901 and 3301 are all prime.
		

Links

• G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 1030031
• The Prime Puzzles and Problems Connection, Puzzle 822

Crossrefs

Cf. A010051, A007090, A107715, A050249.

Programs

Haskell

import Data.List (inits, tails)
a193890 n = a193890_list !! (n-1)
a193890_list = filter f a107715_list where
   f n = (all ((== 1) . a010051) $
               zipWith (\ins (t:tns) -> read $ (ins ++ x3 t ++ tns))
                       (init $ inits $ show n) (init $ tails $ show n))
       where x3 '0' = "0"
             x3 '1' = "3"
             x3 '2' = "6"
             x3 '3' = "9"
-- Reinhard Zumkeller, Aug 11 2011

Maple

S:= NULL:
for x from 2 to 3^10 do
   L:= convert(x, base, 3):
   if numboccur(1,L) mod 3 <> 2 then next fi;
   L1:= subs(2=3,L);
   L2:= subs(1=2,L1);
   for LL in [L1,L2] do
     y:= add(LL[i]*10^(i-1), i=1..nops(L1));
     if isprime(y) then
      good:= true;
      for j from 1 to nops(LL) do
         yp:= y + 2*LL[j]*10^(j-1);
         if not isprime(yp) then
            good:= false;
            break
         fi
      od:
      if good then S:= S, y fi;
     fi;
   od
od:
sort([S]); # Robert Israel, Mar 07 2016

Mathematica

Select[Select[Prime@ Range[10^6], AllTrue[IntegerDigits@ #, MemberQ[{0, 1, 2, 3}, #] &] &], Function[k, AllTrue[Map[FromDigits, Map[MapAt[3 # &, IntegerDigits@ k, #] &, Range@ IntegerLength@ k]], PrimeQ]]] (* Michael De Vlieger, Mar 06 2016, Version 10 *)

Python

from sympy import isprime
from itertools import product
A193890_list = []
for l in range(1,10):
    for d in product('0123',repeat=l):
        p = int(''.join(d))
        if d[0] != '0' and d[-1] in ('1','3') and isprime(p):
            for i in range(len(d)):
                d2 = list(d)
                d2[i] = str(3*int(d[i]))
                if not isprime(int(''.join(d2))):
                    break
            else:
                 A193890_list.append(p) # Chai Wah Wu, Aug 13 2015