A002386 -id:A002386 - cp's OEIS frontend

A201062 Record (maximal) gaps between prime 5-tuples (p, p+4, p+6, p+10, p+12).

90, 1770, 2190, 10080, 24360, 35910, 156750, 208620, 304920, 306390, 328020, 422190, 526350, 639330, 706860, 866460, 1030770, 1111620, 1147440, 1151100, 1447530, 1769670, 1793070, 2024610, 2320170, 2335080, 2403570

Offset: 1

Alexei Kourbatov, Nov 26 2011

nonn

Prime quintuplets (p, p+4, p+6, p+10, p+12) are one of the two types of densest permissible constellations of 5 primes (A022006 and A022007). Average gaps between prime k-tuples can be deduced from the Hardy-Littlewood k-tuple conjecture and are O(log^k(p)), with k=5 for quintuplets. If a gap is larger than all preceding gaps, we call it a maximal gap, or a record gap. Maximal gaps may be significantly larger than average gaps; this sequence suggests that maximal gaps between quintuplets are O(log^6(p)).

A201063 lists initial primes in quintuplets (p, p+4, p+6, p+10, p+12) preceding the maximal gaps. A233433 lists the corresponding primes at the end of the maximal gaps.

			The gap of 90 between quintuplets starting at p=7 and p=97 is the very first gap, so a(1)=90. The gap of 1770 between quintuplets starting at p=97 and p=1867 is a maximal gap - larger than any preceding gap; therefore a(2)=1770. The gap after p=1867 is smaller, so a new term is not added.

Alexei Kourbatov, Table of n, a(n) for n = 1..71
Tony Forbes, Prime k-tuplets
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
Alexei Kourbatov, Maximal gaps between prime 5-tuples (graphs/data up to 10^15)
A. Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.5.2
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Eric W. Weisstein, k-Tuple Conjecture

Cf. A022007 (prime 5-tuples p, p+4, p+6, p+10, p+12), A113274, A113404, A200503, A201596, A201598, A201073, A201051, A201251, A202281, A202361, A201063, A002386, A233433.

DeleteDuplicates[Differences[Select[Partition[Prime[Range[10^7]],5,1],Differences[#]=={4,2,4,2}&][[;;,1]]],GreaterEqual] (* The program generates the first 18 terms of the sequence. *) (* Harvey P. Dale, Apr 20 2025 *)

(1) Upper bound: gaps between prime 5-tuples are smaller than 0.0987*(log p)^6, 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)-0.4), where a = 0.0987*(log p)^5 is the average gap between quintuplets 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 ~0.0987 is reciprocal to the Hardy-Littlewood 5-tuple constant 10.1317...

A201073 Record (maximal) gaps between prime 5-tuples (p, p+2, p+6, p+8, p+12).

Original entry on oeis.org

6, 90, 1380, 14580, 21510, 88830, 97020, 107100, 112140, 301890, 401820, 577710, 689850, 846210, 857010, 986160, 1655130, 2035740, 2266320, 2467290, 2614710, 3305310, 3530220, 3880050, 3885420, 5290440, 5713800, 6049890

Offset: 1

Alexei Kourbatov, Nov 26 2011

nonn

Prime quintuplets (p, p+2, p+6, p+8, p+12) are one of the two types of densest permissible constellations of 5 primes (A022006 and A022007). Average gaps between prime k-tuples can be deduced from the Hardy-Littlewood k-tuple conjecture and are O(log^k(p)), with k=5 for quintuplets. If a gap is larger than any preceding gap, we call it a maximal gap, or a record gap. Maximal gaps may be significantly larger than average gaps; this sequence suggests that maximal gaps are O(log^6(p)).

A201074 lists initial primes in quintuplets (p, p+2, p+6, p+8, p+12) preceding the maximal gaps. A233432 lists the corresponding primes at the end of the maximal gaps.

			The initial four gaps of 6, 90, 1380, 14580 (between quintuplets starting at p=5, 11, 101, 1481, 16061) form an increasing sequence of records. Therefore a(1)=6, a(2)=90, a(3)=1380, and a(4)=14580. The next gap (after 16061) is smaller, so a new term is not added.

Alexei Kourbatov, Table of n, a(n) for n = 1..64
Tony Forbes, Prime k-tuplets
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
Alexei Kourbatov, Maximal gaps between prime 5-tuples (graphs/data up to 10^15)
A. Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242, 2013 and J. Int. Seq. 16 (2013) #13.5.2
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Eric W. Weisstein, k-Tuple Conjecture

Cf. A022006 (prime 5-tuples p, p+2, p+6, p+8, p+12), A113274, A113404, A200503, A201596, A201598, A201051, A201251, A202281, A202361, A201062, A201074, A002386, A233432.

(1) Upper bound: gaps between prime 5-tuples are smaller than 0.0987*(log p)^6, 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)-0.4), where a = 0.0987*(log p)^5 is the average gap between quintuplets 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 ~0.0987 is reciprocal to the Hardy-Littlewood 5-tuple constant 10.1317...

A201251 Record (maximal) gaps between prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20).

Original entry on oeis.org

83160, 195930, 341880, 5414220, 9270030, 18980220, 25622520, 36077370, 51597630, 92184750, 125523090, 140407470, 141896370, 336026460, 403369470, 435390270, 442452570, 627852330, 754383210, 1008582120, 1021464990, 1073692620, 1088148810, 1145336850

Offset: 1

Alexei Kourbatov, Nov 28 2011

Prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20) are one of the two types of densest permissible constellations of 7 primes (A022009 and A022010). Average gaps between prime k-tuples can be deduced from the Hardy-Littlewood k-tuple conjecture and are O(log^k(p)), with k=7 for septuplets. If a gap is larger than any preceding gap, we call it a maximal gap, or a record gap. Maximal gaps may be significantly larger than average gaps; this sequence suggests that maximal gaps are O(log^8(p)).

A201252 lists initial primes in septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20) preceding the maximal gaps. A233038 lists the corresponding primes at the end of the maximal gaps.

			The gap of 83160 between septuplets starting at p=5639 and p=88799 is the very first gap, so a(1)=83160. The gap of 195930 between septuplets starting at p=88799 and p=284729 is a maximal gap - larger than any preceding gap; therefore a(2)=195930. The next gap of 341880 is again a maximal gap, so a(3)=341880. The next gap is smaller, so it does not contribute to the sequence.

Alexei Kourbatov, Table of n, a(n) for n = 1..52
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
Alexei Kourbatov, Maximal gaps between prime k-tuples.
Alexei Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.5.2.
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
Norman Luhn, Patterns of prime k-tuplets & the Hardy-Littlewood constants.
Norman Luhn, Record Gaps Between Prime Septuplets, up to 10^17.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Cf. A022010 (prime septuplets p, p+2, p+8, p+12, p+14, p+18, p+20), A113274, A113404, A200503, A201062, A201073, A201596, A201598, A202281, A202361, A201051, A002386, A233038.

Gaps between prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20) are smaller than 0.02*(log p)^8, where p is the prime at the end of the gap. There is no rigorous proof of this formula. The O(log^8(p)) growth rate is suggested by numerical data and heuristics based on probability considerations.

A082885 Primes followed by a larger-than-average prime gap.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 73, 83, 89, 113, 131, 139, 151, 157, 167, 173, 181, 199, 211, 233, 241, 251, 257, 263, 271, 283, 293, 317, 331, 337, 353, 359, 367, 373, 383, 389, 401, 409, 421, 449, 467, 479, 491, 509, 523

Offset: 1

Labos Elemer, Apr 17 2003

Previous name was: Primes p(j) such that p(j+1)-p(j) > log(p(j)), where log is the natural logarithm.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A082862, A082884, A002386.

Do[s=(Prime[n+1]-Prime[n])/Log[Prime[n]]//N; If[s>1, Print[{n, Prime[n], s}]], {n, 1, 1000}]
Transpose[Select[Partition[Prime[Range[100]],2,1],#[[2]]-#[[1]]> Log[#[[1]]]&]][[1]] (* Harvey P. Dale, Mar 15 2015 *)

p=2;forprime(q=3,1e4,if(q-p>log(p),print1(p", "));p=q) \\ Charles R Greathouse IV, Feb 07 2012

Conjecture: Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = e (A001113). - Alain Rocchelli, Dec 18 2023

New name from Charles R Greathouse IV, Feb 07 2012

A143935 Number of primes between n^K and (n+1)^K, inclusive, where K=log(127)/log(16).

Original entry on oeis.org

2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 5, 1, 4, 2, 3, 3, 4, 1, 5, 3, 3, 4, 3, 3, 3, 4, 4, 3, 5, 4, 3, 5, 2, 4, 5, 4, 5, 5, 3, 5, 5, 2, 6, 5, 4, 4, 4, 5, 5, 7, 5, 5, 3, 5, 6, 3, 8, 3, 4, 5, 6, 7, 5, 6, 8, 5, 4, 6, 6, 3, 7, 5, 4, 8, 5, 8, 6, 3, 7, 7, 6, 8, 7, 4, 5, 6, 5, 9, 9, 7, 6, 6, 6, 6, 7, 6, 4, 8, 5, 8, 8, 4

Offset: 1

T. D. Noe, Sep 05 2008

nonn

This value of K is conjectured to be the least possible such that there is at least one prime in the range n^k and (n+1)^k for all n>0 and k>=K. This value of K was found using exact interval arithmetic. For each n <= 300 and for each prime p in the range n to n^2, we computed an interval k(n,p) such that p is between n^k(n,p) and (n+1)^k(n,p). The intersection of all these intervals produces a list of 29 intervals. The last interval appears to be semi-infinite beginning with K, which is log(127)/log(16). See A143898 for the smallest number in the first interval.
My UBASIC program indicates no prime between 113.457 ... and 126.999 .... Next prime > 113 is 127. I would like someone to check this. - Enoch Haga, Sep 24 2008
It suffices to check members of floor(A002386^(1/k)). - Charles R Greathouse IV, Feb 03 2011
The constant log(127)/log(16) is A194361. - John W. Nicholson, Dec 13 2013

T. D. Noe, Table of n, a(n) for n = 1..10000
Carlos Rivera, Conjecture 60: Generalization of Legendre's Conjecture

Cf. A014085 (number of primes between n^2 and (n+1)^2).

Cf. A144256, A194361.

k= 1.74717117169304146332; Table[Length[Select[Range[Ceiling[n^k],Floor[(n+1)^k]], PrimeQ]], {n,150}]
With[{k=Log[16,127]},Table[Count[Range[Ceiling[n^k],Floor[(n+1)^k]],?PrimeQ],{n,110}]] (* _Harvey P. Dale, Apr 03 2019 *)

Corrected a(15) from 1 to 0 Enoch Haga, Sep 24 2008

My intention was to include the endpoints of the range. Using k=log(127)/log(16), the endpoint for n=15 is exactly 127, which is prime. - T. D. Noe, Sep 25 2008

A205827 Primes prime(k) corresponding to the records in the sequence (prime(k+1)/prime(k))^k.

Original entry on oeis.org

2, 3, 7, 23, 113, 1129, 1327, 19609, 31397, 155921, 360653, 370261, 1357201, 2010733, 17051707, 20831323, 191912783, 436273009, 2300942549, 3842610773, 4302407359, 10726904659, 25056082087, 304599508537, 461690510011, 1346294310749, 1408695493609

Offset: 1

Thomas Ordowski, May 07 2012

Probably A111870 is this sequence with the exception of the term a(4) = 23. - Farideh Firoozbakht, May 07 2012
For n from 5 to 28, a(n) = A111870(n-1). - Donovan Johnson, Oct 26 2012
The statement prime(k) > (prime(k+1)/prime(k))^k for k>=1 is a rewrite of the Firoozbakht conjecture (see link). - John W. Nicholson, Oct 27 2012
Values of k are in A214935.
The logarithmic (base 10) graph seems to be linearly asymptotic to n with slope ~ 1/log(10) which would imply that: log(prime(k)) ~ n as n goes to infinity. [Copy of comment by N. J. A. Sloane, Aug 27 2010 for A111870, copied and corrected for prime(k) by John W. Nicholson, Oct 29 2012]
(prime(k+1)/prime(k))^k ~ e^merit(k), where merit(k) = (prime(k+1)-prime(k))/log(prime(k)). - Thomas Ordowski, Mar 18 2013
Subset of A002386. - John W. Nicholson, Nov 19 2013
Copied comment from A111870 (modified variable to k): (prime(k+1)/prime(k))^k > 1 + merit(k) for k > 2, where merit(k) = (prime(k+1)-prime(k))/log(prime(k)). - Thomas Ordowski, May 14 2012 : Copied and modified by John W. Nicholson, Nov 20 2013

			The sequence (prime(k+1)/prime(k))^k for k=1,2,... starts with:
*1.500, *2.777, 2.744, *6.098, 2.305, 5.001, 2.178, 4.611, *8.054, 1.948, ...,
where records are marked with *. The corresponding primes are a(1)=prime(1)=2, a(2)=prime(2)=3, a(3)=prime(4)=7, a(4)=prime(9)=23, ...

John W. Nicholson, Table of n, a(n) for n = 1..38
Alexei Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015.
Alexei Kourbatov, Upper Bounds for Prime Gaps Related to Firoozbakht’s Conjecture, arXiv:1506.03042 [math.NT], 2015-2019.
Alexei Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2
Wikipedia, Firoozbakht’s conjecture

Cf. A000040, A111870, A214935.

t = {}; p = 2; best = 0; n = 0; While[n++; last = p; p = NextPrime[p]; p <= 100000, f = (p/last)^n; If[f > best, best = f; AppendTo[t, last]]]; t (* T. D. Noe, May 08 2012 *)

record=0;for(n=1,75,current=(A000101[n]/A002386[n]*1.)^A005669[n];if(current>record,record=current;print1(A002386[n],", "))) \\ Each sequence is read in as a vector as to overcome PARI's primelimit. John W. Nicholson, Dec 01 2013

a(n) = A000040(A214935(n)).

a(13)-a(25) from Donovan Johnson, May 08 2012
Definition corrected by Max Alekseyev, Oct 23 2012
Clarified definition with k as index of a(n)=prime(k) instead of index n, John W. Nicholson, Oct 24 2012
a(26)-a(28) from Donovan Johnson, Oct 26 2012
a(29)-a(38) from John W. Nicholson, Dec 01 2013

A079296 Primes ordered by decreasing value of the function p -> sqrt(q) - sqrt(p) where q is the next prime after p.

Original entry on oeis.org

7, 113, 23, 13, 31, 3, 1327, 19, 47, 199, 139, 89, 5, 211, 293, 53, 523, 317, 61, 181, 73, 887, 1129, 83, 37, 241, 2, 43, 283, 1669, 11, 467, 1069, 337, 509, 2477, 131, 2179, 2971, 1259, 773, 1951, 1637, 409, 3271, 421, 151, 1381, 67, 839, 619, 863, 157, 17, 661, 3137

Offset: 1

Thomas Nordhaus, Feb 09 2003

I computed a couple of thousand primes with EXCEL and ordered them accordingly. There is a very small chance that very large prime numbers will change the order of the given terms above.
This sequence only makes sense if the sequence n -> sqrt(p_(n+1)) - sqrt(p_n) is a zero-sequence which is a hard unsolved problem. See also Andrica's conjecture.
For each consecutive prime pair p < q, the number d = sqrt(q) - sqrt(p) is unique. Place d in order from greatest to least and specify p. See Table II in Wolf. A rearrangement of the primes. - Robert G. Wilson v, Oct 18 2012

Donovan Johnson, Table of n, a(n) for n = 1..10000
C. K. Caldwell, Gaps between primes.
Eric W. Weisstein, Andrica's Conjecture
Wikipedia, Andrica's conjecture
Marek Wolf, A note on the Andrica conjecture, arXiv:1010.3945 [math.NT], 2010.

Cf. A078692, A002386, A084974 (records).

lim = 1/5; lst = {}; p = 2; q = 3; While[p < 50000, If[ Sqrt[q] - Sqrt[p] > lim, AppendTo[lst, {p, Sqrt[q] - Sqrt[p]}]]; p = q; q = NextPrime[q]]; First@ Transpose@ Sort[lst, #1[[2]] > #2[[2]] &] (* Robert G. Wilson v, Oct 18 2012 *)

More terms from Robert G. Wilson v, Oct 18 2012

A214935 Index of the primes of A205827, A000720(A205827(n)).

Original entry on oeis.org

1, 2, 4, 9, 30, 189, 217, 2225, 3385, 14357, 30802, 31545, 104071, 149689, 1094421, 1319945, 10655462, 23163298, 112228683, 182837804, 203615628, 486570087, 1094330259, 11992433550, 17883926781, 50070452577, 52302956123, 72178455400

Offset: 1

John W. Nicholson, Oct 28 2012

nonn

A000040(a(n)) = A205827(n).

With pi(x) being the prime counting function, A000720(x), for n from 1 to 3, a(n) = pi(A111870(n)) = A241542(n), for n from 5 to 28, a(n) = pi(A111870(n-1)) = A241542(n-1). - John W. Nicholson, May 10 2014

			a(4) = 9, A000040(9) = 23, and A205827(4) = 23.

John W. Nicholson, Table of n, a(n) for n = 1..38

Cf. A205827.

record=0;i=0;A214935=vectorsmall(38);for(n=1,75,current=(A000101[n]/A002386[n]*1.)^A005669[n];if(current>record,record=current;i++;A214935[i]=A005669[n]))

a(n) = pi(A205827(n)) = A000720(A205827(n)).

a(13)-a(28) from Donovan Johnson, Oct 28 2012

a(29)-a(38) from John W. Nicholson, Dec 01 2013

A008950 Increasing length runs of consecutive composite numbers (starting points).

Original entry on oeis.org

4, 8, 24, 90, 114, 524, 888, 1130, 1328, 9552, 15684, 19610, 31398, 155922, 360654, 370262, 492114, 1349534, 1357202, 2010734, 4652354, 17051708, 20831324, 47326694, 122164748, 189695660, 191912784, 387096134, 436273010, 1294268492

Offset: 1

Mark Cramer (m.cramer(AT)qut.edu.au). Computed by Dennis Yelle (dennis(AT)netcom.com)

There are runs of n consecutive composite numbers for every n. For example, the n numbers (n+1)!+2 ... (n+1)!+n+1 are composite. Such a run may start of course earlier than this. - Joerg Arndt, May 01 2013

Jens Kruse Andersen, Table of n, a(n) for n=1..74 (using A002386)
Jens Kruse Andersen, The Top-20 Prime Gaps
Jens Kruse Andersen, New record prime gap
Jens Kruse Andersen, Maximal gaps
Eric Weisstein's World of Mathematics, Prime Gaps.
J. Young and A. Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.

Cf. A008995, A008996.

maxGap = 1; Reap[Do[p = Prime[n]; gap = Prime[n+1] - p; If[gap > maxGap, Print[p+1]; Sow[p+1]; maxGap = gap], {n, 2, 10^8 }]][[2, 1]] (* Jean-François Alcover, Jun 15 2012 *)

a(n) = A002386(n+1)+1.

a(n) <= (n+1)! + 2. [Joerg Arndt, May 01 2013]

A053695 Differences between record prime gaps.

Original entry on oeis.org

1, 2, 2, 2, 6, 4, 2, 2, 12, 2, 8, 8, 20, 14, 10, 16, 2, 4, 14, 16, 6, 26, 30, 10, 2, 12, 14, 2, 32, 6, 4, 28, 16, 18, 28, 2, 10, 62, 8, 4, 6, 12, 4, 10, 14, 2, 16, 2, 6, 42, 6, 14, 50, 22, 42, 50, 12, 26, 2, 100, 10, 8, 208, 52, 14, 22, 4, 24, 24, 56, 28, 14, 72, 34, 12, 22

Offset: 1

Jeff Burch, Mar 23 2000

The largest known term of this sequence is a(63) = 1132 - 924 = 208. This seems rather strange for a(63) > 2*100+7 where 100 = max {a(k)| k < 63}. {1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,42,50,52,56,62,72,100,208} is the set of the distinct first 75 terms of the sequence. What is the smallest number m such that a(m) = 36? - Farideh Firoozbakht, May 30 2014

Conjecture: a(n) <= A005250(n). Based on the equivalent statement at A005250: A005250(n+1) / A005250(n) <= 2. - John W. Nicholson, Dec 30 2015

John W. Nicholson, Table of n, a(n) for n = 1..81
Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression. International Mathematical Forum, Vol. 13, 2018, No. 2, 65-78, arXiv:1709.05508v3 [math.NT].
Eric Weisstein's World of Mathematics, Prime Gaps
Wikipedia, Prime gap

Cf. A000101, A001223, A002386, A005250, A053686.

m = 2; r = 0; Differences@ Reap[Monitor[Do[If[Set[d, Set[n, NextPrime[m]] - m] > r, Set[r, d]; Sow[d]]; m = n, {i, 10^7}], i]][[-1, -1]] (* Michael De Vlieger, Oct 30 2021 *)

a(n) = A005250(n+1) - A005250(n).

A005250(n+1) = 1 + Sum_{i=1..n} a(i). - John W. Nicholson, Dec 29 2015

Missing term 1 and more terms added by Farideh Firoozbakht, May 30 2014

a(75)-a(76) from John W. Nicholson, Feb 27 2018

cp's OEIS Frontend

A201062 Record (maximal) gaps between prime 5-tuples (p, p+4, p+6, p+10, p+12).

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A201073 Record (maximal) gaps between prime 5-tuples (p, p+2, p+6, p+8, p+12).

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A201251 Record (maximal) gaps between prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20).

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A082885 Primes followed by a larger-than-average prime gap.

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A143935 Number of primes between n^K and (n+1)^K, inclusive, where K=log(127)/log(16).

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A205827 Primes prime(k) corresponding to the records in the sequence (prime(k+1)/prime(k))^k.

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PARI

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A079296 Primes ordered by decreasing value of the function p -> sqrt(q) - sqrt(p) where q is the next prime after p.

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A214935 Index of the primes of A205827, A000720(A205827(n)).

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