Prime index
A000720 is: 2, 3, 4, 6, 9, and 30.
floor( (prime(k)/k)^2 ) is: 2, 2, 3, 4, 6, and 14.
Similarly, ceiling( (prime(k)/k)^2 ) > prime(k+1) - prime(k) holds for all prime(k) < 10^8 with the exception of prime(k) = 7. For prime(k) = 7, 4 = ceiling((prime(k) / k)^2) = prime(k+1) - prime(k).
Stronger than Firoozbakht's conjecture which states that the sequence prime(k)^(1/k) is strictly decreasing.
Conjecture: list is complete. If so, subsequence of
A124147 and
A174635.
Andrew Granville conjectures that lim sup (prime(n+1)-prime(n))/log(prime(n))^2 >= 2/e^gamma = 1.1229189.... If so (or at least if the lim sup is greater than 1) then this sequence is infinite. -
Charles R Greathouse IV, Feb 18 2016
