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A177688
Numbers n such that (n+2)//n - (n+1) is prime, where // represents the concatenation of decimals.
0
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
1,3
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.
EXAMPLE
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.
MATHEMATICA
n2ncQ[n_]:=PrimeQ[FromDigits[Join[IntegerDigits[n+2], IntegerDigits[ n]]]- n-1]; Select[Range[0, 300], n2ncQ] (* Harvey P. Dale, Feb 24 2011 *)
CROSSREFS
KEYWORD
base,easy,nonn
AUTHOR
Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 11 2010
STATUS
approved