A115401 Record differences between prime(n+3) and prime(n). Records in A031165.

5, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 40, 42, 46, 50, 54, 58, 60, 62, 64, 68, 78, 84, 112, 116, 118, 120, 126, 128, 142, 152, 170, 178, 184, 192, 194, 198, 208, 210, 216, 220, 222, 252, 258, 270, 300, 318, 336, 348, 354, 370, 408

Offset: 1

Author

Jonathan Vos Post, Jan 22 2006

Keywords

• easy
• nonn

Comments

This is the k=3 case of the set of sequences "records in a(k,n) = prime(n+k) - prime(n)." The k=1 case is given by A005250 (ncreasing gaps between primes), A000101 [increasing gaps between primes (upper end)] and A002386, which gives lower ends of these gaps. The k=2 case is A031132. The merits of these records are (prime(n+3)-prime(n))/log (prime(n)). The first record merit is 5/log 2 = 16.6096405. The second record merit is 8/log 3 = 16.7672262.

Examples

			a(1) = A031165(1) = prime(4) - prime(1) = 7 - 2 = 5, which is the only odd element of this sequence.
a(2) = A031165(2) = prime(5) - prime(2) = 11 - 3 = 8.
a(3) = A031165(4) = prime(7) - prime(4) = 17 - 7 = 10.
a(4) = A031165(7) = prime(10) - prime(7) = 29 - 17 = 12.
a(5) = A031165(9) = prime(12) - prime(9) = 37 - 23 = 14.
		

Crossrefs

Cf. A000040, A000101, A001223, A002386, A005250, A007530, A031131, A031132, A031165.

Programs

Mathematica

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ@k, k++ ]; k]; d = 0; p = 1; q = 2; r = 3; s = 5; lst = {}; Do[{p, q, r, s} = {q, r, s, NextPrim[s]}; If[s > d + p, d = s - p; AppendTo[lst, d]; Print[d]], {n, 10^8}] (* Robert G. Wilson v *)

Extensions

Corrected and extended by Robert G. Wilson v, Jan 23 2006