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A113274
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Record gaps between twin primes.
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13
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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, 6474, 6552
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| a(n) mod 6 = 0 for each n>0.
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REFERENCES
| Hardy, G. H. and Littlewood, J. E. "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes." Acta Math. 44, 1-70, 1923.
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LINKS
| Richard Fischer, Maximale Lücken (Intervallen) von Primzahlenzwillingen
Alexei Kourbatov, Maximal gaps between prime k-tuples
Luis Rodriguez and Carlos Rivera, Gaps between consecutive twin pairs
Eric W. Weisstein, k-Tuple Conjecture
Eric W. Weisstein, Twin Prime Constant
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FORMULA
| Contribution by Alexei Kourbatov, Dec 29 2011: (Start)
(1) Upper bound: gaps between twin primes are smaller than 0.76*(ln 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(ln(p/a)-1.2), where a = 0.76*(ln p)^2 is the average gap between twin primes near p, as predicted by the Hardy-Littlewood k-tuple 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 Hardy-Littlewood k-tuple conjecture via probability-based heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the k-tuple conjecture itself has no formal proof either). In both formulas, the constant ~0.76 is reciprocal to the twin prime constant 1.32032...
(End)
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EXAMPLE
| The first twin primes are 3,5 and 5,7 so a(0)=5-3=2. The following pair is 11,13 so a(1)=11-5=6. The following pair is 17,19 so 6 remains the record and no terms are added.
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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 *)
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CROSSREFS
| The smallest primes originating the sequence are given in A113275. Cf. A005250.
A036063[k] = a[k+2]-2
Sequence in context: A193764 A159793 A006511 * A181660 A036913 A117311
Adjacent sequences: A113271 A113272 A113273 * A113275 A113276 A113277
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KEYWORD
| nonn
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AUTHOR
| Bernardo Boncompagni (redgolpe(AT)redgolpe.com), Oct 21 2005
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(at)rgwv.com), Oct 22 2005
Corrected terms based on A036063[k]=A113274[k+2]-2, cross-checked with independent computations by Carlos Rivera and Richard Fischer (linked).
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