login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

The smallest prime p such that p//9_(n), its concatenation with n nines, is prime.
1

%I #2 Mar 30 2012 17:40:23

%S 2,5,2,5,5,2,2,7,41,5,41,41,5,23,101,13,17,29,2,37,13,5,5,79,7,19,2,5,

%T 131,47,47,103,59,5,43,89,149,71,17,5,193,29,2,277,73,107,127,79,7,

%U 103,83,19,173,29,2,23,107,103,43,47,5,13,73,503,307,197,83,113,331,163

%N The smallest prime p such that p//9_(n), its concatenation with n nines, is prime.

%C a(n) is the smallest prime p such that 10^n*(p+1)-1 is a prime number.

%C (a) if a(n) = 29 then a(n+1) = 2, (b) if a(n) = 59 and a(n+1) > 2 then a(n+1) = 5, etc.

%C (c) if a(n) = 19 then a(n+1) >= 2, as 1 is not a prime.

%C The smallest prime that is not appearing as such a concatenation seems to be 11:

%C The cases 11//9_(n) with n even, that is 11//9_(2*k), are multiples of 11 and ruled out.

%C Most cases 11//9_(n) with n odd are also ruled out:

%C 11//9_(6*k+1) are multiples of 7. 11//9_(6*k+3) are multiples of 13.

%C It seems where 11//9_(6*k+5) is prime, so is 5//9_(6*k+5).

%D M. du Sautoy: Die Musik der Primzahlen: Auf den Spuren des groessten Raetsels der Mathematik, Deutscher Taschenbuch Verlag, 2006

%D F. Ischebeck: Einladung zur Zahlentheorie, B.I. Wissenschaftsverlag, Mannheim-Leipzig-Wien-Zuerich, 1992

%e 2//9 = 29 = prime(10), 2 = prime(1) is a(1).

%e 5//99 = 599 = prime(109), 5 = prime(3) is a(2).

%e 163//9_(70) is prime, so a(70)=163.

%K base,nonn

%O 1,1

%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 08 2010

%E Used variable n for the index; used more cautious wording for the unproved n=6k+5 case for the 11's - _R. J. Mathar_, May 10 2010