

A182514


Primes prime(n) such that (prime(n+1)/prime(n))^n > n.


6




OFFSET

1,1


COMMENTS

The Firoozbakht conjecture: (prime(n+1))^(1/(n+1)) < prime(n)^(1/n), or prime(n+1) < prime(n)^(1+1/n), prime(n+1)/prime(n) < prime(n)^(1/n), (prime(n+1)/prime(n))^n < prime(n).
Using the Mathematica program shown below, I have found no further terms below 2^27. I conjecture that this sequence is finite and that the terms stated are the only members.  Robert G. Wilson v, May 06 2012 [Warning: this conjecture may be false!  N. J. A. Sloane, Apr 25 2014]
I conjecture the contrary: the sequence is infinite. Note that 10^13 < a(6) <= 1693182318746371.  Charles R Greathouse IV, May 14 2012
[Stronger than Firoozbakht] conjecture: All (prime(n+1)/prime(n))^n values, with n >= 5, are less than n*log(n).  John W. Nicholson, Dec 02 2013, Oct 19 2016
The Firoozbakht conjecture can be rewritten as (log(prime(n+1)) / log(prime(n)))^n < (1+1/n)^n. This suggests the [weaker than Firoozbakht] conjecture: (log(prime(n+1))/log(prime(n)))^n < e.  Daniel Forgues, Apr 26 2014
All a(n) <= a(6) are in A002386, A205827, and A111870.
The inequality in the definition is equivalent to the inequality prime(n+1)prime(n) > log(n)*log(prime(n)) for sufficiently large n.  Thomas Ordowski, Mar 16 2015
Prime indices, A000720(a(n)) = 1, 2, 4, 30, 217, 49749629143526.  John W. Nicholson, Oct 25 2016


REFERENCES

Farhadian, R. (2017). On a New Inequality Related to Consecutive Primes. OECONOMICA, vol 13, pp. 236242.
Farhadian, R. & Jakimczuk. R. (2017). On a New Conjecture of Prime Numbers. Int. Math. Forum, vol. 12, pp. 559  564, https://doi.org/10.12988/imf.2017.7335.
Rivera, C. (2016). Conjecture 78. P_n^((P_n+1/P_n)^n) <= n^P_n, http://www.primepuzzles.net/conjectures/conj_078.htm.


LINKS

Table of n, a(n) for n=1..6.
Reza Farhadian, A New Conjecture On the primes, Preprint, 2016.
Luan Alberto Ferreira, Some consequences of the Firoozbakht's conjecture, arXiv:1604.03496v2 [math.NT], 2016.
A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015.
Nilotpal Kanti Sinha, On a new property of primes that leads to a generalization of Cramer's conjecture, arXiv:1010.1399 [math.NT], 2010.
Wikipedia, Firoozbakhtâ€™s conjecture


EXAMPLE

7 is in the list because, being the 4th prime, and 11 the fifth prime, we verify that (11/7)^4 = 6.09787588507... which is greater than 4.
11 is not on the list because (13/11)^5 = 2.30543740804... and that is less than 5.


MATHEMATICA

Prime[Select[Range[1000], (Prime[# + 1]/Prime[#])^# > # &]] (* Alonso del Arte, May 04 2012 *)
firoozQ[n_, p_, q_] := n * Log[q] > Log[n] + n * Log[p]; k = 1; p = 2; q = 3; While[ k < 2^27, If[ firoozQ[k, p, q], Print[{k, p}]]; k++; p = q; q = NextPrime@ q] (* Robert G. Wilson v, May 06 2012 *)


PROG

(PARI) n=1; p=2; forprime(q=3, 1e6, if((q/p*1.)^n++>n, print1(p", ")); p=q) \\ Charles R Greathouse IV, May 14 2012
(PARI) for(n=1, 75, if((A000101[n]/A002386[n]*1.)^A005669[n]>=A005669[n], print1(A002386[n], ", "))) \\ Each sequence is read in as a vector as to overcome PARI's primelimit \\ John W. Nicholson, Dec 01 2013
(PARI) q=3; n=2; forprime(p=5, 10^9, result=(p/q)^n/(n*log(n)); if(result>1, print(q, " ", p, " ", n, " ", result)); n++; q=p) \\ for stronger than Firoozbakht conjecture \\ John W. Nicholson, Mar 16 2015, Oct 19 2016


CROSSREFS

Cf. A111870.
Sequence in context: A088120 A230778 A111870 * A062935 A083436 A088856
Adjacent sequences: A182511 A182512 A182513 * A182515 A182516 A182517


KEYWORD

nonn


AUTHOR

Thomas Ordowski, May 04 2012


EXTENSIONS

a(6) from John W. Nicholson, Dec 01 2013


STATUS

approved



