This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A182514 #138 Apr 02 2019 04:06:22
%S A182514 2,3,7,113,1327,1693182318746371
%N A182514 Primes prime(n) such that (prime(n+1)/prime(n))^n > n.
%C A182514 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).
%C A182514 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]
%C A182514 I conjecture the contrary: the sequence is infinite. Note that 10^13 < a(6) <= 1693182318746371. - _Charles R Greathouse IV_, May 14 2012
%C A182514 [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
%C A182514 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
%C A182514 All a(n) <= a(6) are in A002386, A205827, and A111870.
%C A182514 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
%C A182514 Prime indices, A000720(a(n)) = 1, 2, 4, 30, 217, 49749629143526. - _John W. Nicholson_, Oct 25 2016
%D A182514 Farhadian, R. (2017). On a New Inequality Related to Consecutive Primes. OECONOMICA, vol 13, pp. 236-242.
%H A182514 Reza Farhadian, <a href="http://www.primepuzzles.net/conjectures/Reza%20Faradian%20Conjecture.pdf">A New Conjecture On the primes</a>, Preprint, 2016.
%H A182514 R. Farhadian, and R. Jakimczuk, <a href="https://doi.org/10.12988/imf.2017.7335">On a New Conjecture of Prime Numbers</a> Int. Math. Forum, vol. 12, 2017, pp. 559-564.
%H A182514 Luan Alberto Ferreira, <a href="http://arxiv.org/abs/1604.03496">Some consequences of the Firoozbakht's conjecture</a>, arXiv:1604.03496v2 [math.NT], 2016.
%H A182514 Luan Alberto Ferreira, Hugo Luiz Mariano, <a href="https://doi.org/10.1007/s40863-018-0113-0">Prime gaps and the Firoozbakht Conjecture</a>, São Paulo J. Math. Sci. (2018), 1-11.
%H A182514 A. Kourbatov, <a href="http://arxiv.org/abs/1503.01744">Verification of the Firoozbakht conjecture for primes up to four quintillion</a>, arXiv:1503.01744 [math.NT], 2015.
%H A182514 A. Kourbatov, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Kourbatov/kourb7.html">Upper bounds for prime gaps related to Firoozbakht's conjecture</a>, J. Int. Seq. 18 (2015) 15.11.2.
%H A182514 Carlos Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_078.htm">Conjecture 78. P_n^((P_n+1/P_n)^n) <= n^P_n</a>, 2016.
%H A182514 Nilotpal Kanti Sinha, <a href="http://arxiv.org/abs/1010.1399">On a new property of primes that leads to a generalization of Cramer's conjecture</a>, arXiv:1010.1399 [math.NT], 2010.
%H A182514 Matt Visser, <a href="https://arxiv.org/abs/1904.00499">Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap</a>, arXiv:1904.00499 [math.NT], 2019.
%H A182514 Wikipedia, <a href="http://en.wikipedia.org/wiki/Firoozbakht%E2%80%99s_conjecture">Firoozbakht’s conjecture</a>
%e A182514 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.
%e A182514 11 is not on the list because (13/11)^5 = 2.30543740804... and that is less than 5.
%t A182514 Prime[Select[Range[1000], (Prime[# + 1]/Prime[#])^# > # &]] (* _Alonso del Arte_, May 04 2012 *)
%t A182514 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 *)
%o A182514 (PARI) 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
%o A182514 (PARI) 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
%o A182514 (PARI) 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
%Y A182514 Cf. A111870.
%K A182514 nonn
%O A182514 1,1
%A A182514 _Thomas Ordowski_, May 04 2012
%E A182514 a(6) from _John W. Nicholson_, Dec 01 2013