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Search: id:A115401
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| A115401 |
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Record differences between prime(n+3) and prime(n). Records in A115400. |
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+0
2
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| 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; listen)
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OFFSET
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1,1
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COMMENT
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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.
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FORMULA
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Record values in A115400 = Record values of prime(n+3) - prime(n). Record values of A000040(n+3) - A000040(n).
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EXAMPLE
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a(1) = A115400(1) = prime(4) - prime(1) = 7 - 2 = 5, which is the only odd element of this sequence.
a(2) = A115400(2) = prime(5) - prime(2) = 11 - 3 = 8.
a(3) = A115400(4) = prime(7) - prime(4) = 17 - 7 = 10.
a(4) = A115400(7) = prime(10) - prime(7) = 29 - 17 = 12.
a(5) = A115400(9) = prime(12) - prime(9) = 37 - 23 = 14.
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A000040, A000101, A001223, A002386, A005250, A007530, A031131, A031132, A115400.
Sequence in context: A121901 A049195 A064362 this_sequence A065528 A050936 A084146
Adjacent sequences: A115398 A115399 A115400 this_sequence A115402 A115403 A115404
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post (jvospost2(AT)yahoo.com), Jan 22 2006
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EXTENSIONS
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Corrected and extended by Robert G. Wilson v (rgwv(at)rgwv.com), Jan 23 2006
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