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

%I A205827 #97 Aug 24 2025 05:50:22
%S A205827 2,3,7,23,113,1129,1327,19609,31397,155921,360653,370261,1357201,
%T A205827 2010733,17051707,20831323,191912783,436273009,2300942549,3842610773,
%U A205827 4302407359,10726904659,25056082087,304599508537,461690510011,1346294310749,1408695493609
%N A205827 Primes prime(k) corresponding to the records in the sequence (prime(k+1)/prime(k))^k.
%C A205827 Probably A111870 is this sequence with the exception of the term a(4) = 23. - _Farideh Firoozbakht_, May 07 2012
%C A205827 For n from 5 to 28, a(n) = A111870(n-1). - _Donovan Johnson_, Oct 26 2012
%C A205827 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
%C A205827 Values of k are in A214935.
%C A205827 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]
%C A205827 (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
%C A205827 Subset of A002386. - _John W. Nicholson_, Nov 19 2013
%C A205827 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
%H A205827 John W. Nicholson, <a href="/A205827/b205827.txt">Table of n, a(n) for n = 1..38</a>
%H A205827 Alexei 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 A205827 Alexei Kourbatov, <a href="https://arxiv.org/abs/1506.03042">Upper Bounds for Prime Gaps Related to Firoozbakht’s Conjecture</a>, arXiv:1506.03042 [math.NT], 2015-2019.
%H A205827 Alexei 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 A205827 Wikipedia, <a href="http://en.wikipedia.org/wiki/Firoozbakht%E2%80%99s_conjecture">Firoozbakht’s conjecture</a>
%F A205827 a(n) = A000040(A214935(n)).
%e A205827 The sequence (prime(k+1)/prime(k))^k for k=1,2,... starts with:
%e A205827 *1.500, *2.777, 2.744, *6.098, 2.305, 5.001, 2.178, 4.611, *8.054, 1.948, ...,
%e A205827 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, ...
%t A205827 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 *)
%o A205827 (PARI) 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
%Y A205827 Cf. A000040, A111870, A214935.
%K A205827 nonn,changed
%O A205827 1,1
%A A205827 _Thomas Ordowski_, May 07 2012
%E A205827 a(13)-a(25) from _Donovan Johnson_, May 08 2012
%E A205827 Definition corrected by _Max Alekseyev_, Oct 23 2012
%E A205827 Clarified definition with k as index of a(n)=prime(k) instead of index n, _John W. Nicholson_, Oct 24 2012
%E A205827 a(26)-a(28) from _Donovan Johnson_, Oct 26 2012
%E A205827 a(29)-a(38) from _John W. Nicholson_, Dec 01 2013