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

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

Author

Thomas Ordowski, May 07 2012

Keywords

• nonn
• changed

Comments

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

Examples

			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, ...
		

Links

• 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

Crossrefs

Cf. A000040, A111870, A214935.

Programs

Mathematica

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 *)

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

Formula

a(n) = A000040(A214935(n)).

Extensions

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