

A113274


Record gaps between twin primes.


18



2, 6, 12, 18, 30, 36, 72, 150, 168, 210, 282, 372, 498, 630, 924, 930, 1008, 1452, 1512, 1530, 1722, 1902, 2190, 2256, 2832, 2868, 3012, 3102, 3180, 3480, 3804, 4770, 5292, 6030, 6282, 6474, 6552, 6648, 7050, 7980, 8040, 8994, 9312, 9318, 10200, 10338, 10668
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OFFSET

1,1


COMMENTS

a(n) mod 6 = 0 for each n>1.


LINKS



FORMULA

(1) Upper bound: gaps between twin primes are smaller than 0.76*(log p)^3, where p is the prime at the end of the gap.
(2) Estimate for the actual size of the maximal gap that ends at p: maximal gap = a(log(p/a)1.2), where a = 0.76*(log p)^2 is the average gap between twin primes near p, as predicted by the HardyLittlewood ktuple conjecture.
Formulas (1) and (2) are asymptotically equal as p tends to infinity. However, (1) yields values greater than all known gaps, while (2) yields "good guesses" that may be either above or below the actual size of known maximal gaps.
Both formulas (1) and (2) are derived from the HardyLittlewood ktuple conjecture via probabilitybased heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the ktuple conjecture itself has no formal proof either). In both formulas, the constant ~0.76 is reciprocal to the twin prime constant 1.32032...
(End)


EXAMPLE

The first twin primes are 3,5 and 5,7 so a(0)=53=2. The following pair is 11,13 so a(1)=115=6. The following pair is 17,19 so 6 remains the record and no terms are added.


MATHEMATICA

NextLowerTwinPrim[n_] := Block[{k = n + 6}, While[ !PrimeQ[k]  !PrimeQ[k + 2], k+=6]; k]; p = 5; r = 2; t = {2}; Do[ q = NextLowerTwinPrim[p]; If[q > r + p, AppendTo[t, q  p]; Print[{p, q  p}]; r = q  p]; p = q, {n, 10^9}]; t (* Robert G. Wilson v, Oct 22 2005 *)
DeleteDuplicates[Differences[Select[Partition[Prime[Range[10^7]], 2, 1], #[[2]]#[[1]] == 2&][[All, 2]]], GreaterEqual] (* The program generates the first 27 terms of the sequence. *) (* Harvey P. Dale, Dec 31 2022 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

Corrected terms based on A036063, crosschecked with independent computations by Carlos Rivera and Richard Fischer (linked).
Terms up to a(72) are given in Kourbatov (2013), terms up to a(75) in Oliveira e Silva website.


STATUS

approved



