A174031 The smallest integer k>0 such that the double-concatenation prime(n) // prime(n+1) // k is a prime number.

3, 3, 1, 19, 1, 1, 1, 1, 1, 1, 9, 11, 11, 17, 3, 1, 1, 3, 11, 17, 21, 19, 1, 7, 37, 7, 23, 37, 7, 1, 7, 7, 7, 11, 7, 33, 29, 31, 1, 13, 11, 17, 7, 11, 11, 9, 9, 1, 7, 7, 1, 13, 11, 19, 67, 1, 13, 21, 49, 13, 13, 1, 1, 23, 1, 1, 29, 1, 29, 7

Offset: 1

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 06 2010

Keywords

• base
• nonn

Comments

Leading zeros in k are not allowed.

All entries k are odd with final digit 1, 3, 7 or 9.

Dirichlet's prime number theorem for arithmetic progressions says that the sequence is infinite.

Conjecture: 1 appears infinitely often.

Examples

			n=1: 2//3//1 = 231 = 3 * 7 * 11 is not prime, so k<>1. 233 = prime(51), therefore 3 is the first entry.
n=2: 3//5//1 = 351 = 3^3 * 13 is not prime, so k <> 1, but 353 = prime(71), therefore 3 is the second entry.
n=30: 113//127//1 = 1131271 = prime(87976), so the 30th entry is 1.
		

Crossrefs

Cf. A030461, A030459, A030469, A171154

Programs

Maple

read("transforms") ;
A174031 := proc(n) for e from 1 do if isprime(digcatL([ithprime(n),ithprime(n+1),e])) then return e ; end if; end do: end proc:

Extensions

Entries checked; replaced variables by OEIS standard names - R. J. Mathar, Nov 17 2010