OFFSET
1,2
COMMENTS
Firoozbakht's conjecture: prime(n+1) < prime(n)^(1+1/n).
Firoozbakht's conjecture restated for this sequence: a(n) >= n.
I further conjecture that n = 1,2,4 are the only values of n with a(n) = n.
Record values of a(n) occur when prime(n) and prime(n+1) are twin primes.
Upper bound for all n: a(n) < (1/2)*(prime(n)+2)*log(prime(n)).
REFERENCES
Paulo Ribenboim, The little book of bigger primes, 2nd edition, Springer, 2004, p. 185.
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, arXiv:1506.03042 [math.NT], 2015.
Carlos Rivera, Conjecture 30
FORMULA
a(n) = floor(log(prime(n))/(log(prime(n+1)) - log(prime(n)))).
EXAMPLE
prime(1)=2; a(1)=1 because k=1 is the largest k for which 3 < 2^(1+1/k).
prime(2)=3; a(2)=2 because k=2 is the largest k for which 5 < 3^(1+1/k).
prime(10)=29; a(10)=50 because k=50 is the largest k for which 31 < 29^(1+1/k).
MATHEMATICA
Table[Floor[Log@ Prime@ n /(Log@ Prime[n + 1] - Log@ Prime@ n)], {n, 58}] (* Michael De Vlieger, Oct 08 2015 *)
PROG
(Magma) [Floor(Log(NthPrime(n))/(Log(NthPrime(n+1))-Log(NthPrime(n)))): n in [1..60]]; // Vincenzo Librandi, Oct 08 2015
(PARI) a(n) = floor(log(prime(n))/(log(prime(n+1)) - log(prime(n)))) \\ Michel Marcus, Oct 10 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Alexei Kourbatov, Oct 08 2015
EXTENSIONS
More terms from Vincenzo Librandi, Oct 08 2015
STATUS
approved