OFFSET
1,1
COMMENTS
Firoozbakht's conjecture, prime(n+1) < prime(n)^(1 + 1/n), is equivalent to a(n) > prime(n). See also A182134.
Here prime(n) = A000040(n). The conjecture is also equivalent to a(n) - prime(n) >= A001223(n), the n-th gap between primes. See also A246778(n) = floor(prime(n)^(1 + 1/n)) - prime(n).
It is also conjectured that the equality a(n) - prime(n) = A001223(n) holds only for n in the set {1, 2, 3, 4, 8}, see A246782. a(n) is also largest prime less than prime(n)^(1 + 1/n), since prime(n)^(1 + 1/n) is never prime. - Farideh Firoozbakht, Nov 03 2014
LINKS
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:1010.1399 [math.NT]
Wikipedia, Firoozbakht's conjecture
FORMULA
MAPLE
seq(prevprime(ceil(ithprime(n)^(1+1/n))), n=1..100); # Robert Israel, Nov 03 2014
MATHEMATICA
Table[NextPrime[Prime[n]^(1 + 1/n), -1], {n, 64}] (* Farideh Firoozbakht, Nov 03 2014 *)
PROG
(PARI) a(n)=precprime(prime(n)^(1+1/n))
(PARI) a(n)=precprime(sqrtnint(prime(n)^(n+1), n)) \\ Charles R Greathouse IV, Oct 29 2018
(Haskell)
a245396 n = a244365 n (a182134 n) -- Reinhard Zumkeller, Nov 16 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Nov 03 2014
STATUS
approved