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 A263023 #25 Sep 08 2022 08:46:14
%S A263023 1,2,4,4,14,9,25,15,13,50,19,35,77,42,32,37,122,43,72,153,54,88,63,52,
%T A263023 113,235,121,252,130,40,156,108,339,71,375,128,134,210,144,151,466,96,
%U A263023 504,256,523,90,96,304,618,313,214,657,134,233,240,247,755,255
%N A263023 Largest integer k such that prime(n+1) < prime(n)^(1+1/k).
%C A263023 Firoozbakht's conjecture: prime(n+1) < prime(n)^(1+1/n).
%C A263023 Firoozbakht's conjecture restated for this sequence: a(n) >= n.
%C A263023 I further conjecture that n = 1,2,4 are the only values of n with a(n) = n.
%C A263023 Record values of a(n) occur when prime(n) and prime(n+1) are twin primes.
%C A263023 Upper bound for all n: a(n) < (1/2)*(prime(n)+2)*log(prime(n)).
%D A263023 Paulo Ribenboim, The little book of bigger primes, 2nd edition, Springer, 2004, p. 185.
%H A263023 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 A263023 A. Kourbatov, <a href="http://arxiv.org/abs/1506.03042">Upper bounds for prime gaps related to Firoozbakht's conjecture</a>, arXiv:1506.03042 [math.NT], 2015.
%H A263023 Carlos Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_030.htm">Conjecture 30</a>
%F A263023 a(n) = floor(log(prime(n))/(log(prime(n+1)) - log(prime(n)))).
%e A263023 prime(1)=2; a(1)=1 because k=1 is the largest k for which 3 < 2^(1+1/k).
%e A263023 prime(2)=3; a(2)=2 because k=2 is the largest k for which 5 < 3^(1+1/k).
%e A263023 prime(10)=29; a(10)=50 because k=50 is the largest k for which 31 < 29^(1+1/k).
%t A263023 Table[Floor[Log@ Prime@ n /(Log@ Prime[n + 1] - Log@ Prime@ n)], {n, 58}] (* _Michael De Vlieger_, Oct 08 2015 *)
%o A263023 (Magma) [Floor(Log(NthPrime(n))/(Log(NthPrime(n+1))-Log(NthPrime(n)))): n in [1..60]]; // _Vincenzo Librandi_, Oct 08 2015
%o A263023 (PARI) a(n) = floor(log(prime(n))/(log(prime(n+1)) - log(prime(n)))) \\ _Michel Marcus_, Oct 10 2015
%Y A263023 Cf. A000040, A002386, A182514, A182519, A245396, A246776, A246778, A249669.
%K A263023 nonn
%O A263023 1,2
%A A263023 _Alexei Kourbatov_, Oct 08 2015
%E A263023 More terms from _Vincenzo Librandi_, Oct 08 2015