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 A115401 Record differences between prime(n+3) and prime(n). Records in A031165. 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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. LINKS EXAMPLE 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. 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 *) CROSSREFS Cf. A000040, A000101, A001223, A002386, A005250, A007530, A031131, A031132, A031165. Sequence in context: A064362 A173298 A248356 * A314377 A314378 A314379 Adjacent sequences:  A115398 A115399 A115400 * A115402 A115403 A115404 KEYWORD easy,nonn AUTHOR Jonathan Vos Post, Jan 22 2006 EXTENSIONS Corrected and extended by Robert G. Wilson v, Jan 23 2006 STATUS approved

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Last modified October 22 22:22 EDT 2020. Contains 337962 sequences. (Running on oeis4.)