A177688 Numbers n such that (n+2)//n - (n+1) is prime, where // represents the concatenation of decimals.

0, 1, 4, 6, 7, 9, 12, 13, 15, 18, 19, 22, 25, 28, 31, 33, 39, 46, 48, 49, 52, 60, 61, 64, 67, 73, 75, 84, 85, 88, 90, 99, 100, 103, 106, 132, 133, 135, 136, 138, 142, 156, 160, 163, 171, 178, 181, 183, 187, 190, 198, 201, 202, 211, 220, 222, 229, 238, 241, 246, 252

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 11 2010

Keywords

• base
• easy
• nonn

Comments

If n is a k-digit number, then we demand that p = (n+2) * 10^k + n - (n+1) is a prime number, obviously of the form p = (n+2) * 10^k - 1, so the decimal representation of p is n+1 followed by k times the digit 9.

The sequence is infinite, proof with Dirichlet's prime number (in arithmetic progressions) theorem.

Note that numbers of the form (n+2)//n + (n+1) are multiples of 3 and do not generate primes.

Examples

			2//0 - 1 = 20 - 1 = 19 = prime(8), 0 is first term;
3//1 - 2 = 31 - 2 = 29 = prime(10), 1 is 2nd term;
6//4 - 5 = 64 - 5 = 59 = prime(17), 4 is 3rd term.
		

Crossrefs

Cf. A000040, A065582, A095995, A173976, A177435.

Programs

Mathematica

n2ncQ[n_]:=PrimeQ[FromDigits[Join[IntegerDigits[n+2], IntegerDigits[ n]]]- n-1]; Select[Range[0,300],n2ncQ]  (* Harvey P. Dale, Feb 24 2011 *)