A002386 -id:A002386 - cp's OEIS frontend

A075051 Smallest prime for which the n closest primes are smaller.

3, 113, 113, 113, 1327, 1327, 15683, 15683, 248909, 265621, 492113, 492113, 3851459, 7743233, 18640103, 18640103, 18640103, 435917249, 435917249, 435917249, 649580171, 649580171, 19187736221, 19187736221, 19187736221, 94746870541, 94746870541, 673420121333, 1975675658371

Offset: 1

Neil Fernandez, Oct 10 2002

nonn

It is surprising that few of the above entries are at the beginning of a prime gap in A000230 or A002386.

			The smallest prime number for which the three closest primes to itself are all smaller than itself is 113 (the closest primes being 109, 107 and 103). So a(3)=113.

Cf. A001223, A074979, A074982, A075030, A075037, A075038, A075043, A075050.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; k = 1; Do[ps = Table[0, {n + 1}]; ps = Append[ps, Max[k, 1]]; While[ps = Drop[ps, 1]; ps = Append[ps, NextPrim[ ps[[ -1]]]]; ps[[ -1]] - ps[[ -2]] <= ps[[ -2]] - ps[[1]], ]; Print[ ps[[ -2]]]; k = PrevPrim[ ps[[1]]], {n, 1, 30}]

Edited and extended by Robert G. Wilson v, Oct 12 2002

a(23)-a(29) from Donovan Johnson, Jun 19 2008

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

A246782 Numbers k such that A182134(k)=2, i.e., there exist only two primes p with prime(k) < p < prime(k)^(1+1/k).

Original entry on oeis.org

5, 6, 7, 9, 10, 11, 14, 15, 22, 23, 28, 29, 30, 45, 46, 61, 66, 216, 217, 367, 3793, 1319945, 1576499, 8040877, 17567976, 44405858, 445538764, 1478061204, 3643075047, 17440041685, 190836014732, 714573709895, 714573709896

Offset: 1

Farideh Firoozbakht, Oct 12 2014

Firoozbakht's conjecture says that for every n, there exists at least one prime p such that prime(n) < p < prime(n)^(1+1/n).
Let A(m) = {n | A182134(n) = m} where A182134(n) = #{p | p is prime and prime(n) < p < prime(n)^(1+1/n)}. This sequence gives the terms of A(2) and the sequence A246781 gives the terms of A(3).
The only known indices n for which A182134(n) = 1 are {1, 2, 3, 4, 8}.  It is conjectured that this is the complete set A(1).
Conjecture: For all m, where m is greater than one, A(m) is an infinite set.
a1 = 49749629143524, a2 = 1475067052906944 and a3 = 1475067052906945 are three large terms of the sequence. It is interesting that a3 - a2 = 1.
Conjecture: The sequence is infinite.
Next term is greater than 25000000.
a(34) > 10^12. - Robert Price, Nov 01 2014
The conjecture that A(1)={1, 2, 3, 4, 8} holds through 10^12. - Robert Price, Nov 01 2014

			5 is in the sequence since there exists only two primes p, prime(5) < p < prime(5)^(1+1/5). Note that prime(5) = 11, 11^(1+1/5) ~ 17.77 and 11 < 13 < 17 < 17.77.

A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015
Wikipedia, Firoozbakht's conjecture

Cf. A000040, A002386, A005669, A182134, A246781, A249566.

a246782 n = a246782_list !! (n-1)
a246782_list = filter ((== 2) . a182134) [1..]
-- Reinhard Zumkeller, Nov 17 2014

np[n_]:=(a = Prime[n]; b = a^(1 + 1/n); Length[Select[Range[a+1,b], PrimeQ]]); Do[If[np[n] == 2,Print[n]], {n, 25000000}]

for(n=1,oo,2==primepi(prime(n)^(1+1/n))-n&&print1(n", ")) \\ M. F. Hasler, Nov 03 2014

a(26)-a(27) from Robert Price, Oct 24 2014

a(28)-a(33) from Robert Price, Nov 01 2014

A002540 Increasing gaps between prime-powers.

Original entry on oeis.org

1, 5, 13, 19, 32, 53, 89, 139, 199, 293, 887, 1129, 1331, 5591, 8467, 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

List of prime-powers where A057820 increases.

The entry K=a(k) is the start of the smallest chain of m=A121492(k) consecutive numbers such that lcm(1,2,...,K) = lcm(1,2,...,K,K+1) = lcm(1,2,...,K,K+1,K+2) = ... = lcm(1,2,...,K,...,K+m-1). See A121493. - Lekraj Beedassy, Aug 03 2006

J. P. Gram, Undersoegelser angaaende maengden af primtal under en given graense, Det Kongelige Danskevidenskabernes Selskabs Skrifter, series 6, vol. 2 (1884), 183-288; see p. 255.
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).

Jan Kristian Haugland, Table of n, a(n) for n = 1..87 (terms 1..79 from Donovan Johnson). The extra terms are copied from A002386 as the associated prime gaps do not contain any prime powers.
J. P. Gram, Undersoegelser angaaende maengden af primtal under en given graense (1884) [Scanned copy of page 255 with annotations by Victor Meally and N. J. A. Sloane]
Des MacHale and Joseph Manning, Maximal runs of strictly composite integers, The Mathematical Gazette, Vol. 99, No. 545 (2015), pp. 213-219.
Victor Meally, Letter to N. J. A. Sloane, Mar 17, 1980.
Index entries for primes, gaps between.

Cf. A000961 (prime-powers), A057820 (gaps), A002386 (prime equivalent), A094158, A121493.

s = {}; gap = 0; p1 = 1; Do[If[PrimePowerQ[p2], If[(d = p2 - p1) > gap, gap = d; AppendTo[s, p1]]; p1 = p2], {p2, 2, 10^6}]; s (* Amiram Eldar, Dec 12 2022 *)
Join[{1},Rest[Module[{nn=5*10^6,pps},pps=Select[Range[nn],PrimePowerQ]; DeleteDuplicates[ Thread[{Most[ pps],Differences[ pps]}],GreaterEqual[ #1[[2]],#2[[2]]]&]][[;;,1]]]] (* The program generates the first 27 terms of the sequence. *) (* Harvey P. Dale, Aug 20 2024 *)

/* calculates smaller terms - see Donovan Johnson link for larger terms */
isA000961(n) = (omega(n) == 1 || n == 1)
d_max=0;n_prev=1;for(n=2,1e6,if(isA000961(n),d=n-n_prev;if(d>d_max,print(n_prev);d_max=d);n_prev=n)) \\ Michael B. Porter, Oct 31 2009

Second term corrected by Donovan Johnson, Nov 13 2008 (cf. A094158)

a(28)-a(79) from Donovan Johnson, Nov 14 2008

A036061 Increasing gaps among twin primes: the largest prime of the starting twin pair.

Original entry on oeis.org

5, 7, 19, 43, 73, 313, 349, 661, 2383, 5881, 13399, 18541, 24421, 62299, 187909, 687523, 688453, 850351, 2868961, 4869913, 9923989, 14656519, 17382481, 30752233, 32822371, 96894043, 136283431, 234966931, 248641039, 255949951, 390817729, 698542489, 2466641071

Offset: 1

N. J. A. Sloane

nonn

Has many terms in common with A054691, but neither of the two is a subsequence of the other one. - M. F. Hasler, May 07 2022

Martin Raab, Table of n, a(n) for n = 1..82
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Tomás Oliveira e Silva, Gaps between twin primes
Randall Rathbun, Twin Prime Gaps, NMBRTHRY Mailing List, Nov 23 1998.
Eric Weisstein's World of Mathematics, Prime Constellation

Cf. A036062, A036063, A002386, A113275.

Block[{s = Select[Partition[Prime@ Range[10^7], 2, 1], Subtract @@ # == -2 &][[All, -1]], t}, t = Differences@ s; Map[s[[FirstPosition[t, #]]] &, Union@ FoldList[Max, t]][[All, 1]]] (* Michael De Vlieger, Jan 18 2019 *)

a(n) = A036062(n) - A036063(n).

a(n) = A113275(n) + 2.

Terms 5, 7 prepended by Max Alekseyev, Nov 05 2015

a(17) corrected and a(31)-a(33) from Sean A. Irvine, Oct 21 2020

A062529 Smallest prime p such that there is a gap of 2^n between p and the next prime.

Original entry on oeis.org

2, 3, 7, 89, 1831, 5591, 89689, 3851459, 1872851947, 1999066711391, 22790428875364879

Offset: 0

Labos Elemer, Jun 25 2001

nonn

a(11) <= 79419801290172271035479303914142441 and  a(12) <= 55128448018333565337014555712123010955456071077000028555991469751. - Abhiram R Devesh, Aug 09 2014
From Zhining Yang, Dec 02 2022: (Start)
a(11) = 5333419265419188034369535864125349, 34 digits, discovered by Helmut Spielauer in 2013
a(12) = 55128448018333565337014555712123010955456071077000028555991469751, 65 digits, discovered by Helmut Spielauer in 2013
a(13) = 192180552346991956641101827551986346298837407139466361414211497406670710665021150917759713696699494356609164354068319457039591759, 129 digits, discovered by Dana Jacobsen in 2016
a(14) = 267552521*631#/210 - 9606, 268 digits, discovered by Dana Jacobsen in 2016
a(15) = 2717*1303#/268590 - 16670, 552 digits, discovered by Dana Jacobsen in 2014
a(16) = 7079*3559#/9870 - 36310, 1517 digits, discovered by Michiel Jansen, Pierre Cami, and Jens Kruse Andersen in 2013
a(17) = 1111111111111111111*9059#/(11#*5237) - 86522, 3899 digits, discovered by Hans Rosenthal in 2017
a(11) to a(17) were searched from Thomas R. Nicely's homepage. (End)
Importantly, the values in the previous comment are only upper bounds on a(11)-a(17), and are (almost certainly) not the correct values. As of this comment, the largest prime gap length whose first occurrence is known is 1676 < 2^11. - Brian Kehrig, May 01 2025

			a(2)=7 because 7 and 11 are consecutive primes with difference 2^2=4.
a(3)=89 because 89 and 97 are consecutive primes with difference 2^3=8.

Cino Hilliard, TwinPrimes Java code.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101].
Thomas R. Nicely, Other tables of prime gaps

Cf. A000230, A062530, A101232, A002386.

f[n_] := Block[{k = 1}, While[Prime[k + 1] != n + Prime[k], k++ ]; Prime[k]]; Do[ Print[ f[2^n]], {n, 0, 10}] (* Robert G. Wilson v, Jan 13 2005 *)

import sympy
n=0
while n>=0:
    p=2
    while sympy.nextprime(p)-p!=(2**n):
        p=sympy.nextprime(p)
    print(p)
    n=n+1
    p=sympy.nextprime(p)
## Abhiram R Devesh, Aug 09 2014

a(n) = A000230(2^(n-1)). - R. J. Mathar, Jan 12 2007

a(n) = A000230(2^(n-1)) = Min{p|nextprime(p)-p = 2^n} [may need adjusting since offset has been changed].

a(10) sent by Robert G. Wilson v, Jan 13 2005

a(11)-a(12) removed by Brian Kehrig, May 01 2025

A113275 Lesser of twin primes for which the gap before the following twin primes is a record.

Original entry on oeis.org

3, 5, 17, 41, 71, 311, 347, 659, 2381, 5879, 13397, 18539, 24419, 62297, 187907, 687521, 688451, 850349, 2868959, 4869911, 9923987, 14656517, 17382479, 30752231, 32822369, 96894041, 136283429, 234966929, 248641037, 255949949

Offset: 1

Bernardo Boncompagni, Oct 21 2005

nonn

			The smallest twin prime pair is 3, 5, then 5, 7 so a(1) = 3; the following pair is 11, 13 so a(2) = 5 because 11 - 5 = 6 > 5 - 3 = 2; the following pair is 17, 19: since 17 - 11 = 6 = 11 - 5 nothing happens; the following pair is 29, 31 so a(3)= 17 because 29 - 17 = 12 > 11 - 5 = 6.

Martin Raab, Table of n, a(n) for n = 1..82 (first 75 terms from Max Alekseyev)
Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.
Alexei Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013.
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Mersenneforum, Gaps between prime pairs (Twin Primes).
Tomás Oliveira e Silva, Gaps between twin primes

Record gaps are given in A113274. Cf. A002386.

NextLowerTwinPrim[n_] := Block[{k = n + 2}, While[ !PrimeQ[k] || !PrimeQ[k + 2], k++ ]; k]; p = 3; r = 0; t = {3}; Do[q = NextLowerTwinPrim[p]; If[q > r + p, AppendTo[t, p]; r = q - p]; p = q, {n, 10^9}] (* Robert G. Wilson v, Oct 22 2005 *)

a(n) = A036061(n) - 2.

a(n) = A036062(n) - A113274(n).

a(22)-a(30) from Robert G. Wilson v, Oct 22 2005

Terms up to a(72) are listed in Kourbatov (2013), terms up to a(75) in Oliveira e Silva's website, added by Max Alekseyev, Nov 06 2015

A127596 Numbers k such that 1 + Sum_{i=1..k-1} A001223(i)*(-1)^i = 0.

Original entry on oeis.org

2, 4, 14, 22, 28, 233, 249, 261, 488, 497, 511, 515, 519, 526, 531, 534, 548, 562, 620, 633, 635, 2985, 3119, 3123, 3128, 3157, 4350, 4358, 4392, 4438, 4474, 4484, 4606, 4610, 4759, 5191, 12493, 1761067, 2785124, 2785152, 2785718, 2785729, 2867471

Offset: 1

Manuel Valdivia, Apr 03 2007

nonn

Or, with prime(0) = 1, numbers k such that Sum_{i=0..k-1} (prime(i+1)-prime(i))*(-1)^i = Sum_{i=0..k-1} (A008578(i+1)-A008578(i))*(-1)^i = 0.

There are 313 terms < 10^7, 846 terms < 10^8, 1161 terms < 10^9.

			1 - A001223(1) = 1 - 1 = 0, hence 2 is a term.
1 - A001223(1) + A001223(2) - A001223(3) = 1 - 1 + 2 - 2 = 0, hence 4 is a term.

Jinyuan Wang, Table of n, a(n) for n = 1..1161 (first 846 terms from Klaus Brockhaus)
Eric Weisstein's World of Mathematics, Andrica's Conjecture
Eric Weisstein's World of Mathematics, Prime Difference Function

Cf. A001223 (differences between consecutive primes), A008578 (prime numbers at the beginning of the 20th century), A000101 (increasing gaps between primes, upper end), A002386 (increasing gaps between primes, lower end).

Cf. A282178 (prime(a(n))), A330545, A330547.

S=0; Do[j=Prime[n+1]; i=Prime[n]; d[n]=j-i; S=S+(d[n]*(-1)^n); If[S+1==0, Print[Table[j|PrimePi[j]|S+1]]], {n,1,10^7,1}]

{m=10^8; n=1; p=1; e=1; s=0; while(nKlaus Brockhaus, Apr 29 2007 */

Edited by Klaus Brockhaus, Apr 29 2007

A141857 Primes congruent to 10 mod 11.

Original entry on oeis.org

43, 109, 131, 197, 241, 263, 307, 373, 439, 461, 571, 593, 659, 769, 857, 967, 1033, 1187, 1231, 1297, 1319, 1429, 1451, 1583, 1627, 1693, 1759, 1847, 1913, 1979, 2089, 2111, 2221, 2243, 2287, 2309, 2441, 2551, 2617, 2683, 2749, 2837, 2903, 2969, 3079, 3167

Offset: 1

N. J. A. Sloane, Jul 11 2008

From N. J. A. Sloane, Jan 29 2014: (Start)
If p = 11k+10 then k must be odd, so p is also of the form 22r+21.
Note that the differences are surely unbounded - compare A002386. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002386.

[ p: p in PrimesUpTo(5000) | p mod 11 eq 10 ]; // Vincenzo Librandi, Apr 19 2011

Select[Range[10, 50000, 11], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Jun 13 2011 *)
Select[Range[43, 50000, 22], PrimeQ] (* Zak Seidov, Jan 29 2014 *)
Select[Prime[Range[500]],Mod[#,11]==10&] (* Harvey P. Dale, Jan 10 2023 *)

is(n)=isprime(n) && n%11==10 \\ Charles R Greathouse IV, Jul 02 2016

a(n) ~ 10n log n. - Charles R Greathouse IV, Jul 02 2016

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

A075051 Smallest prime for which the n closest primes are smaller.

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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.

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A246782 Numbers k such that A182134(k)=2, i.e., there exist only two primes p with prime(k) < p < prime(k)^(1+1/k).

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A002540 Increasing gaps between prime-powers.

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A036061 Increasing gaps among twin primes: the largest prime of the starting twin pair.

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A062529 Smallest prime p such that there is a gap of 2^n between p and the next prime.

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A113275 Lesser of twin primes for which the gap before the following twin primes is a record.

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