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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 June 7 11:34 EDT 2023. Contains 363157 sequences. (Running on oeis4.)