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

Author

Amarnath Murthy, Mar 30 2002

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A069488 and A069489.

Cf. A010051, A038618.

Programs

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

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 *)

Extensions

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