A090534 Least n-digit prime in which every two-digit string is also a prime, or 0 if no such number exists. (n-1 two-digit string primes occur.)

2, 11, 113, 1117, 11113, 111119, 1111379, 11111117, 111111113, 1111111973, 11111111113, 111111111379, 1111111111717, 11111111111719, 111111111111713, 1111111111111319, 11111111111111119, 111111111111111131

Offset: 1

Author

Amarnath Murthy, Dec 08 2003

Keywords

• base
• nonn

Comments

Conjecture: sequence is infinite. Notes: (1) If n==1 (mod 3) then in most cases a(n) is a permutation of (n-1), 1's and a 7, else a(n) is a digit permutation of (n-1) ones and a 3; (2) Obviously all the digits except the most significant one must be odd.

The least potential candidate, lpc(n), is of the form (10^n-1)/9. The following is a(n)-lpc(n): 1, 0, 2, 6, 2, 8, 268, 6, 2, 862, 2, 268, 606, 608, 602, 208, 8, 20, 0, 20, 68, 600, 0, 2, 606, .... - Robert G. Wilson v, May 12 2014

Examples

			a(5) = 11113 in which 11,11,11,13 are the two-digit string primes.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A242377.

Programs

Mathematica

fQ[p_] := Block[{id = IntegerDigits@ p}, Union@ PrimeQ[ FromDigits@# & /@ Partition[id, 2, 1]] == {True}]; f[n_] := Block[{p = NextPrime[(10^n - 1)/9 - 1]}, While[! fQ@ p, p = NextPrime@ p]; p]; f[1] = 2; Array[ f, 19] (* Robert G. Wilson v, May 12 2014 *)

Extensions

More terms from David Wasserman, Dec 21 2005