A069490 Primes > 1000 in which every substring of lengths 2 and 3 are also prime.

1373, 3137, 3797, 6131, 6173, 6197, 9719, 11311, 11317, 17971, 31379, 61379, 71971, 113131, 113173, 113797, 131311, 131317, 131797, 179719, 317971, 431311, 431797, 617971, 1131131, 1131379, 1311311, 1313797, 1317971, 3131137, 3131311

Offset

1,1

Comments

For all terms: substrings of length 3 correspond to one of the first 21 terms of A069489. - Reinhard Zumkeller, Jun 08 2015

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..100

Example

11317 is a term as the substrings of length 2, i.e., 11, 13, 31, 17 and the three substrings of length 3, i.e., 113, 131 and 317 are all prime.

Mathematica

Do[ If[ Union[ PrimeQ[ Map[ FromDigits, Partition[ IntegerDigits[ Prime[n]], 2, 1]]]] == Union[ PrimeQ[ Map[ FromDigits, Partition[ IntegerDigits[ Prime[n]], 3, 1]]]] == {True}, Print[ Prime[n]]], {n, PrimePi[1000] + 1, 10^5}]
 Select[Prime[Range[169, 226000]], AllTrue[FromDigits/@Flatten[Table[ Partition[ IntegerDigits[ #], k, 1], {k, {2, 3}}], 1], PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 02 2021 *)

Prog

(Haskell)
import Data.Set (fromList, deleteFindMin, union)
a069490 n = a069490_list !! (n-1)
a069490_list = f $ fromList [1..9] where
   f s | m < 1000               = f s''
       | h m && a010051' m == 1 = m : f s''
       | otherwise              = f s''
       where s'' = union s' $ fromList $ map (+ (m * 10)) [1, 3, 7, 9]
             (m, s') = deleteFindMin s
   h x = x < 100 && a010051' x == 1 ||
         a010051' (x `mod` 1000) == 1 &&
         a010051' (x `mod` 100) == 1 && h (x `div` 10)
-- Reinhard Zumkeller, Jun 08 2015

Crossrefs

Cf. A069488 and A069489.

Cf. A010051, A038618.

Sequence in context: A140125 A179915 A168167 * A239974 A258964 A349235

Adjacent sequences: A069487 A069488 A069489 * A069491 A069492 A069493

Keyword

nonn

Author

Amarnath Murthy, Mar 30 2002

Extensions

Edited, corrected and extended by Robert G. Wilson v, Apr 12 2002

Status

approved