A111870 -id:A111870 - cp's OEIS frontend

A111871 Prime gaps q-p with n-th record merit referred to in A111870.

1, 2, 4, 14, 22, 34, 52, 72, 86, 96, 112, 132, 148, 180, 210, 248, 282, 320, 336, 354, 382, 456, 514, 532, 582, 588, 602, 652, 716, 766, 906, 1132, 1328, 1356, 1370, 1442, 1476, 1572

Offset: 1

N. J. A. Sloane, based on correspondence with Ed Pegg Jr, Nov 23 2005

The prime gaps q-p (corresponding to a(n)=p in A111870) have increasing record merit (q-p)/log(p). However, the prime gaps themselves are almost always monotonically increasing (with very high probability), but not always! And we do have an exception in the list above: a(14)=148 < a(13)=154! (But see next comment!)

Because the erroneous A111870(13) = 4652353 term was removed, a(13) = 154 was removed. This sequence is therefore monotonically increasing. - John W. Nicholson, Nov 18 2013

			A111870(4) = 113 and the next larger prime is 127, so 127 - A111870(4) = a(4) = 14.

Ed Pegg, Jr., Posting to Seq Fan mailing list, Nov 23, 2005

Jens Kruse Andersen, Norman Luhn, Record prime gaps.

For the primes p corresponding to the prime gaps q-p with n-th record merit, see A111870.

Cf. A002386, A277552.

a(n) = A277552(n) - A111870(n). - Bobby Jacobs, Nov 13 2016

Corrected and edited by Daniel Forgues, Nov 11 2009 and Nov 20 2009
Because the erroneous A111870(13) = 4652353 term was removed, a(13) = 154 was removed by John W. Nicholson, Nov 18 2013
a(33)-a(35) inserted by Bobby Jacobs, Nov 08 2016
a(37) added by Bobby Jacobs, Nov 09 2016
a(38) added by Rodolfo Ruiz-Huidobro, May 14 2024

A241542 Indices (i.e., value of A000720 = primepi) of primes in A111870.

Original entry on oeis.org

1, 2, 4, 30, 189, 217, 2225, 3385, 14357, 30802, 31545, 104071, 149689, 1094421, 1319945, 10655462, 23163298, 112228683, 182837804, 203615628, 486570087, 1094330259, 11992433550, 17883926781, 50070452577, 52302956123, 72178455400, 94906079600, 473258870471, 662221289043

Offset: 1

M. F. Hasler, Apr 25 2014

nonn

Amiram Eldar, Table of n, a(n) for n = 1..32 (calculated using Kim Walisch's primecount)
Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000720, A111870.

a(n) = A000720(A111870(n)).

a(20)-a(30) from Amiram Eldar, Sep 03 2024

A002386 Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k.

Original entry on oeis.org

2, 3, 7, 23, 89, 113, 523, 887, 1129, 1327, 9551, 15683, 19609, 31397, 155921, 360653, 370261, 492113, 1349533, 1357201, 2010733, 4652353, 17051707, 20831323, 47326693, 122164747, 189695659, 191912783, 387096133, 436273009, 1294268491

Offset: 1

N. J. A. Sloane

See the links by Jens Kruse Andersen et al. for very large gaps.

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 133.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 6.1, Table 1.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 14.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Brian Kehrig, Table of n, a(n) for n = 1..83 (first 75 terms from M. F. Hasler and N. J. A. Sloane, terms n = 76..77 added by Charles R Greathouse IV)
R. K. Guy, Letter to N. J. A. Sloane, Aug 1986
R. K. Guy, Letter to N. J. A. Sloane, 1987
Lutz Kämmerer, A fast probabilistic component-by-component construction of exactly integrating rank-1 lattices and applications, arXiv:2012.14263 [math.NA], 2020.
Jens Kruse Andersen and Norman Luhn, Record Prime Gaps
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, arXiv:1506.03042 [math.NT], 2015; and J. Int. Seq. 18 (2015) #15.11.2.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Thomas R. Nicely, New maximal prime gaps and first occurrences, Math. Comput. 68,227 (1999) 1311-1315.
Tomás Oliveira e Silva, Gaps between consecutive primes
D. Shanks, On maximal gaps between successive primes, Math. Comp., 18 (1964), 646-651.
Matt Visser, Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap, arXiv:1904.00499 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Prime Gaps
Wikipedia, Prime gap
Robert G. Wilson v, Notes (no date)
J. Young and A. Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.
Index entries for primes, gaps between

Cf. A000040, A001223, A000101 (upper ends), A005250 (record gaps), A000230, A111870, A111943.
Cf. A005669, A134266, A070866.
See also A205827(n) = A000040(A214935(n)), A182514(n) = A000040(A241540(n)).

s = {2}; gm = 1; Do[p = Prime[n]; g = Prime[n + 1] - p; If[g > gm, Print[p]; AppendTo[s, p]; gm = g], {n, 2, 1000000}]; s   (* Jean-François Alcover, Mar 31 2011 *)
Module[{nn=10^7,pr,df},pr=Prime[Range[nn]];df=Differences[pr];DeleteDuplicates[ Thread[ {Most[ pr],df}],GreaterEqual[#1[[2]],#2[[2]]]&]][[All,1]] (* The program generates the first 26 terms of the sequence. *) (* Harvey P. Dale, Sep 24 2022 *)

a(n)=local(p,g);if(n<2,2*(n>0),p=a(n-1);g=nextprime(p+1)-p;while(p=nextprime(p+1),if(nextprime(p+1)-p>g,break));p) /* Michael Somos, Feb 07 2004 */

p=q=2;g=0;until( g<(q=nextprime(1+p=q))-p && print1(q-g=q-p,","),) \\ M. F. Hasler, Dec 13 2007

a(n) = A000101(n) - A005250(n) = A008950(n-1) - 1. - M. F. Hasler, Dec 13 2007
A000720(a(n)) = A005669(n).
a(n) = A000040(A005669(n)). - M. F. Hasler, Apr 26 2014

Definition clarified by Harvey P. Dale, Sep 24 2022

A114403 Triprime gaps. First differences of A014612.

Original entry on oeis.org

4, 6, 2, 7, 1, 2, 12, 2, 1, 5, 2, 11, 3, 2, 2, 5, 1, 2, 14, 6, 1, 3, 3, 5, 4, 2, 1, 7, 1, 5, 8, 9, 1, 5, 1, 10, 1, 5, 1, 1, 2, 1, 7, 4, 2, 2, 5, 12, 5, 10, 8, 1, 5, 2, 4, 2, 1, 1, 9, 3, 3, 5, 2, 5, 2, 4, 3, 2, 1, 1, 4, 2, 18, 6, 2, 4, 3, 7, 1, 5, 5, 2, 9, 2, 1

Offset: 1

Jonathan Vos Post, Nov 25 2005

			a(1) = 4 = 12-8 where 8 is the first triprime and 12 is the second.
a(2) = 6 = 18-12
a(3) = 2 = 20-18
a(4) = 7 = 27-20

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A014612, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

is3Alm := proc(n::integer) local ifa,ex,i ; ifa := op(2,ifactors(n)) ; ex := 0 ; for i from 1 to nops(ifa) do ex := ex+ op(2,op(i,ifa)) ; od : if ex = 3 then RETURN(true) ; else RETURN(false) ; fi ; end: A014612 := proc(n::integer) local resul,i; i :=1; resul := 8 ; while i < n do resul := resul + 1 ; if is3Alm(resul) then i := i+1 ; fi ; od ; RETURN(resul) ; end: A114403 := proc(n::integer) RETURN(A014612(n+1)-A014612(n)) ; end: for n from 1 to 160 do printf("%d,",A114403(n)) ; od: # R. J. Mathar, Apr 25 2006

Differences[Select[Range[425], PrimeOmega[#] == 3 &]] (* Jayanta Basu, Jul 01 2013 *)

a(n) = A014612(n+1) - A014612(n).

Corrected and extended by R. J. Mathar, Apr 25 2006

A114412 Records in semiprime gaps ordered by merit.

Original entry on oeis.org

2, 3, 4, 6, 11, 19, 24, 28, 30, 32, 38, 47, 54, 70, 74, 107, 110, 112, 120, 126, 146

Offset: 1

Jonathan Vos Post, Nov 25 2005

There is an associated index list n = 1, 2, 4, 6, 34, 422, 1765, 4585, 8112, 8650, 8861, 75150, ... and an associated semiprime list A001358(n) = 4, 6, 10, 15, 1418, 6559, 17965, 32777, 35103, 35981, 340894, ... - R. J. Mathar, Mar 15 2009

			Records defined in terms of A065516 and A001358:
.
  n  A065516(n)  A065516(n)/log_10(A001358(n))
  =  ==========  ==============================
  1       2      2 / log_10(4)  = 3.32192809...
  2       3      3 / log_10(6)  = 3.85529162...
  3       1      1 / log_10(9)  = 1.04795163...
  4       4      4 / log_10(10) = 4.00000000
  5       1      1 / log_10(14) = 0.87250286...
  6       6      6 / log_10(15) = 5.10164492...
  7       1      1 / log_10(21) = 0.75630419...
  8       3      3 / log_10(22) = 2.23476557...
  9       1      1 / log_10(25) = 0.71533827...

Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Prime Gaps.

Cf. A001358, A065516, A111870, A111871, A113688-A113693, A114403-A114411, A114412-A114422.

sp = 4; m0 = 0;  l = {}; lim = 1000000;
For[i = 5, i <= lim, i++, If[PrimeOmega[i] == 2, m = (i - sp)/Log[sp]; If[m > m0, m0 = m; AppendTo[l, i - sp]]; sp = i] ]; l (* Robert Price, Oct 29 2018 *)

a(n) = records in A065516(n)/log_10(A001358(n)) = records in (A001358(n+1) - A001358(n))/log_10(A001358(n)).

Corrected and extended by Charles R Greathouse IV, Oct 05 2006

a(16)-a(21) from Donovan Johnson, Feb 17 2010

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

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

A082891 Smallest prime p such that q = (r-p)/log(p) > n, where r is the next prime after p.

Original entry on oeis.org

2, 7, 1129, 1327, 19609, 31397, 155921, 370261, 1357201, 2010733, 20831323, 20831323, 191912783, 436273009, 3842610773, 10726904659, 25056082087, 25056082087, 25056082087, 1346294310749, 1408695493609, 2614941710599, 13829048559701, 19581334192423, 19581334192423

Offset: 1

Labos Elemer, Apr 17 2003

nonn

Is lim superior(q(n)) = +infinity? See A082892.

			For n = 11 and 12: k = 1319945: p(k+1) = 20831533, p(k) = 20831323, d = p(k+1) - p(k) = 210, log(20831321) = 16.852..., q = 210/16.852... = 12.4615... > 12 and also > 11 for the first time, so a(11) = a(12) = 20831323.

Amiram Eldar, Table of n, a(n) for n = 1..32 (calculated using the data at A111870)

Cf. A082862, A082884, A082885, A082886, A082888, A082889, A082890, A082892, A111870.

Do[s=(Prime[n+1]-Prime[n])/Log[Prime[n]]//N; If[s>11, Print[{n, Prime[n], Prime[n+1], s, Log[Prime[n]]//N}]], {n, 1000000, 100000000}]

lista(pmax) = {my(n = 1, prv = 2, d, m); print1(2, ", "); forprime(p=3, pmax, d = p-prv; m = floor(d/log(prv)); if(m > n, for(k = 1, m-n, print1(prv, ", ")); n = m); prv=p);} \\ Amiram Eldar, Nov 04 2024

a(n)= Min{p(x); (p(x+1)-p(x))/log(p(x)) > n}.

a(10) corrected and a(13)-a(25) added by Amiram Eldar, Nov 04 2024

A111943 Prime p with prime gap q - p of n-th record Cramer-Shanks-Granville ratio, where q is smallest prime larger than p and C-S-G ratio is (q-p)/(log p)^2.

Original entry on oeis.org

23, 113, 1327, 31397, 370261, 2010733, 20831323, 25056082087, 2614941710599, 19581334192423, 218209405436543, 1693182318746371

Offset: 1

N. J. A. Sloane, following emails from R. K. Guy and Ed Pegg Jr, Nov 27 2005

Primes less than 23 are anomalous and are excluded.
a(12) was discovered by Bertil Nyman in 1999.
Shanks conjectures that the ratio will never reach 1. Granville conjectures the opposite: that the ratio will exceed or come arbitrarily close to 2/e^gamma = 1.1229....
Firoozbakht's conjecture implies that the ratio is below 1-1/log(p) for all primes p>=11; see Th.1 of arXiv:1506.03042. In Cramér's probabilistic model of primes, the ratio is below 1-1/log(p) for almost all maximal gaps between primes; see A235402. - Alexei Kourbatov, Jan 28 2016

			-----------------------------
n   ratio                a(n)
-----------------------------
1   0.6103                23
2   0.6264               113
3   0.6575              1327
4   0.6715             31397
5   0.6812            370261
6   0.7025           2010733
7   0.7394          20831323
8   0.7953       25056082087
9   0.7975     2614941710599
10  0.8177    19581334192423
11  0.8311   218209405436543
12  0.9206  1693182318746371

R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, A8.

Andrew Granville, Harald Cramér and the distribution of prime numbers, Scandinavian Actuarial J. 1 (1995), pp. 12-28.
Alexei Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, arXiv:1506.03042 [math.NT], 2015; J. Integer Sequences, 18 (2015), Article 15.11.2.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Daniel Shanks, On maximal gaps between successive primes, Math. Comp. 18 (88) (1964), 646-651.
Eric Weisstein's World of Mathematics, Prime Gaps.
Eric Weisstein's World of Mathematics, Cramer-Granville Conjecture.
Eric Weisstein's World of Mathematics, Shanks Conjecture (and Wolf Conjecture).

Subsequence of A002386.

Cf. A111870, A166363.

r=CSG=0;p=13;forprime(q=17,1e8,if(q-p>r,r=q-p; t=r/log(p)^2; if(t>CSG, CSG=t; print1(p", ")));p=q) \\ Charles R Greathouse IV, Apr 07 2013

Corrected and edited (p_n could be misinterpreted as the n-th prime) by Daniel Forgues, Nov 20 2009

Edited by Charles R Greathouse IV, May 14 2010

A182514 Primes prime(n) such that (prime(n+1)/prime(n))^n > n.

Original entry on oeis.org

2, 3, 7, 113, 1327, 1693182318746371

Offset: 1

Thomas Ordowski, May 04 2012

nonn

The Firoozbakht conjecture: (prime(n+1))^(1/(n+1)) < prime(n)^(1/n), or prime(n+1) < prime(n)^(1+1/n), prime(n+1)/prime(n) < prime(n)^(1/n), (prime(n+1)/prime(n))^n < prime(n).
Using the Mathematica program shown below, I have found no further terms below 2^27. I conjecture that this sequence is finite and that the terms stated are the only members. - Robert G. Wilson v, May 06 2012 [Warning: this conjecture may be false! - N. J. A. Sloane, Apr 25 2014]
I conjecture the contrary: the sequence is infinite. Note that 10^13 < a(6) <= 1693182318746371. - Charles R Greathouse IV, May 14 2012
[Stronger than Firoozbakht] conjecture: All (prime(n+1)/prime(n))^n values, with n >= 5, are less than n*log(n). - John W. Nicholson, Dec 02 2013, Oct 19 2016
The Firoozbakht conjecture can be rewritten as (log(prime(n+1)) / log(prime(n)))^n < (1+1/n)^n. This suggests the [weaker than Firoozbakht] conjecture: (log(prime(n+1))/log(prime(n)))^n < e. - Daniel Forgues, Apr 26 2014
All a(n) <= a(6) are in A002386, A205827, and A111870.
The inequality in the definition is equivalent to the inequality prime(n+1)-prime(n) > log(n)*log(prime(n)) for sufficiently large n. - Thomas Ordowski, Mar 16 2015
Prime indices, A000720(a(n)) = 1, 2, 4, 30, 217, 49749629143526. - John W. Nicholson, Oct 25 2016

			7 is in the list because, being the 4th prime, and 11 the fifth prime, we verify that (11/7)^4 = 6.09787588507... which is greater than 4.
11 is not on the list because (13/11)^5 = 2.30543740804... and that is less than 5.

Farhadian, R. (2017). On a New Inequality Related to Consecutive Primes. OECONOMICA, vol 13, pp. 236-242.

Reza Farhadian, A New Conjecture On the primes, Preprint, 2016.
R. Farhadian, and R. Jakimczuk, On a New Conjecture of Prime Numbers Int. Math. Forum, vol. 12, 2017, pp. 559-564.
Luan Alberto Ferreira, Some consequences of the Firoozbakht's conjecture, arXiv:1604.03496v2 [math.NT], 2016.
Luan Alberto Ferreira, Hugo Luiz Mariano, Prime gaps and the Firoozbakht Conjecture, São Paulo J. Math. Sci. (2018), 1-11.
A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015.
A. Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2.
Carlos Rivera, Conjecture 78. P_n^((P_n+1/P_n)^n) <= n^P_n, 2016.
Nilotpal Kanti Sinha, On a new property of primes that leads to a generalization of Cramer's conjecture, arXiv:1010.1399 [math.NT], 2010.
Matt Visser, Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap, arXiv:1904.00499 [math.NT], 2019.
Wikipedia, Firoozbakht’s conjecture

Cf. A111870.

Prime[Select[Range[1000], (Prime[# + 1]/Prime[#])^# > # &]] (* Alonso del Arte, May 04 2012 *)
firoozQ[n_, p_, q_] := n * Log[q] > Log[n] + n * Log[p]; k = 1; p = 2; q = 3; While[ k < 2^27, If[ firoozQ[k, p, q], Print[{k, p}]]; k++; p = q; q = NextPrime@ q] (* Robert G. Wilson v, May 06 2012 *)

n=1;p=2;forprime(q=3,1e6,if((q/p*1.)^n++>n, print1(p", "));p=q) \\ Charles R Greathouse IV, May 14 2012

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

q=3;n=2; forprime(p=5, 10^9,result=(p/q)^n/(n*log(n));if(result>1, print(q," ",p, " ", n, " ", result));n++;q=p) \\ for stronger than Firoozbakht conjecture \\ John W. Nicholson, Mar 16 2015, Oct 19 2016

a(6) from John W. Nicholson, Dec 01 2013

cp's OEIS Frontend

A111871 Prime gaps q-p with n-th record merit referred to in A111870.

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A241542 Indices (i.e., value of A000720 = primepi) of primes in A111870.

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A002386 Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k.

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A114403 Triprime gaps. First differences of A014612.

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Maple

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A114412 Records in semiprime gaps ordered by merit.

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

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