A115401 Record differences between prime(n+3) and prime(n). Differences 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

1,1

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

Table of n, a(n) for n=1..57.

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 *)
DeleteDuplicates[#[[4]]-#[[1]]&/@Partition[Prime[Range[10^7]], 4, 1], GreaterEqual] (* The program generates the first 53 terms of the sequence. *) (* Harvey P. Dale, Jan 31 2026 *)

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

Definition clarified by Harvey P. Dale, Jan 31 2026

Status

approved