A157772 Numbers n such that 100n + 13 is prime.

0, 1, 3, 6, 10, 12, 16, 19, 21, 22, 27, 33, 34, 36, 40, 45, 48, 51, 54, 58, 61, 70, 72, 85, 87, 90, 94, 96, 103, 105, 106, 111, 112, 118, 121, 124, 126, 127, 133, 135, 136, 139, 147, 148, 150, 153, 154, 159, 177, 180, 183, 184, 187, 189, 190, 192, 198, 199, 201, 210, 213, 216

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 06 2009

Keywords

• nonn
• easy

Comments

The sequence is infinite, because by Dirichlet's theorem there are infinitely many primes in the arithmetic sequence A*n+B (n=1,2,...) if A an B are relatively prime.

The sequence also has an infinite set of pairs a(k+1)=a(k)+1 (two consecutive naturals), but no set of three consecutive naturals (each third natural is divisible by 3)

No term of the sequence is of form 3k+2, because the sum of digits of 100*(3k+2)+13 is divisible by 3, violating the requirement of the definition.

Indices (as k-th prime) of the first members are 6, 30, 65, 112, 170, 198, 255, 293, 319, 330, 396, 466, 480, 505, 554, 612, 648, 684, 714, 763, 797, 902, 922, 1061, 1086, 1121, 1164, 1186, 1265, 1286, 1295, ...

Examples

			a(1)=0: 100*0+13=13 smallest prime which ends in 13, see A000040(6).
a(2)=1: 100*1+13=113 second prime which ends in 13, see A000040(30).
		

Crossrefs

Cf. A088262 (6th row). - R. J. Mathar, Apr 18 2009

Programs

PARI

isok(n) = isprime(100*n + 13) \\ Michel Marcus, Jul 22 2013

Formula

{a(n): 100*a(n)+13 in A000040}.

Extensions

Edited, 27 inserted by R. J. Mathar, Apr 18 2009