

A205827


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


10



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
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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(n1).  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 Firoozbakhtâ€™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, Firoozbakhtâ€™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



