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
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
KEYWORD
nonn,base,hard,more
AUTHOR
Arkadiusz Wesolowski, Aug 08 2011
STATUS
approved