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

1,1

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

Links

John W. Nicholson, Table of n, a(n) for n = 1..38

A. 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 preprint, 2015.

A. Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2

Wikipedia, Firoozbakht’s conjecture

Formula

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

Example

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, ...

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 *)

Prog

(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

Crossrefs

Cf. A000040, A111870, A214935.

Sequence in context: A077213 A112601 A181609 * A098544 A176706 A281529

Adjacent sequences: A205824 A205825 A205826 * A205828 A205829 A205830

Keyword

nonn

Author

Thomas Ordowski, May 07 2012

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

Status

approved