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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). 3
11, 311, 1301, 10133, 1030031 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

LINKS

Table of n, a(n) for n=1..5.

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 1030031

The Prime Puzzles and Problems Connection, Puzzle 822

EXAMPLE

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

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 *)

PROG

(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

(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 is_prime(int(''.join(d2))):

                    break

            else:

                 A193890_list.append(p) # Chai Wah Wu, Aug 13 2015

CROSSREFS

Cf. A010051, A007090, A107715, A050249.

Sequence in context: A250551 A001280 A100445 * A317744 A185071 A060495

Adjacent sequences:  A193887 A193888 A193889 * A193891 A193892 A193893

KEYWORD

nonn,base,hard,more

AUTHOR

Arkadiusz Wesolowski, Aug 08 2011

STATUS

approved

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Last modified September 26 23:42 EDT 2021. Contains 347673 sequences. (Running on oeis4.)