

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
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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 *)


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



