OFFSET
1,2
COMMENTS
Firoozbakht's conjecture says that for every n, there exists at least one prime p where, prime(n) < p < prime(n)^(1 + 1/n). Hence if Firoozbakht's conjecture is true, then there is no m such that np(m) = 0.
Conjecture: For every positive integer n, a(n) exists.
a(65) > 10^12. - Robert Price, Nov 12 2014
LINKS
Robert Price, Table of n, a(n) for n = 1..64
A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015
A. Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2
Nilotpal Kanti Sinha, On a new property of primes that leads to a generalization of Cramer's conjecture, arXiv:1010.1399v2 [math.NT], 2010.
Wikipedia, Firoozbakht's conjecture.
EXAMPLE
a(6) = 55 since the number of primes p such that prime(55) < p < prime(55)^(1 + 1/55) is 6 and 55 is the smallest number with this property.
MATHEMATICA
np[n_]:=(b=Prime[n]; Length[Select[Range[b+1, b^(1 + 1/n)], PrimeQ]]); a[n_]:=(For[m=1, np[m] !=n, m++]; m);
Do[Print[a[n]], {n, 37}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Farideh Firoozbakht and Jahangeer Kholdi, Oct 10 2014
STATUS
approved