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A113274 Record gaps between twin primes. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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.

LINKS

Alexei Kourbatov, Table of n, a(n) for n = 1..72 [Entered by N. J. A. Sloane from A. Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053, 2013]

Richard Fischer, Maximale Lücken (Intervallen) von Primzahlenzwillingen

Alexei Kourbatov, Maximal gaps between prime k-tuples

A. Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242, 2013.

Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053, 2013.

Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959, 2014

Tomás Oliveira e Silva, Gaps between twin primes

Luis Rodriguez and Carlos Rivera, Gaps between consecutive twin pairs

Eric W. Weisstein, k-Tuple Conjecture

Eric W. Weisstein, Twin Prime Constant

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)

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.

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 *)

CROSSREFS

The smallest primes originating the sequence are given in A113275. Cf. A008407, A005250, A002386.

A036063[k] = a[k+2]-2

Sequence in context: A159793 A238739 A006511 * A181660 A036913 A117311

Adjacent sequences:  A113271 A113272 A113273 * A113275 A113276 A113277

KEYWORD

nonn

AUTHOR

Bernardo Boncompagni, Oct 21 2005

EXTENSIONS

More terms from Robert G. Wilson v, 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).

STATUS

approved

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Last modified October 20 21:27 EDT 2014. Contains 248371 sequences.