OFFSET
1,1
COMMENTS
Andrica's conjecture states that sqrt(prime(n+1))-sqrt(prime(n)) < 1 for all n. Since equality cannot happen, this is equivalent to say that all terms of is sequence are >= 1.
Sequence A074976 is based on the same idea (rounding to the nearest integer instead).
It is a remarkable coincidence(?) that very often, especially around "peaks", a symmetric pattern "x, y, x" occurs: 2, 7, 2,... 10, 5, 10,... 13, 2, 13,... 20, 4, 20, ..., 11, 5, 11, ...
LINKS
M. F. Hasler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Andrica's conjecture
EXAMPLE
a(1) = floor(1/(sqrt(3) - sqrt(2))) = floor(1/(1.73-1.41)) = floor(1/0.32) = floor(3.15) = 3.
a(2) = floor(1/(sqrt(5) - sqrt(3))) = floor(1/(2.236-1.732)) = floor(1/0.504) = floor(1.98) = 1.
PROG
(PARI) a(n)=1\(sqrt(prime(n+1))-sqrt(prime(n))) \\ M. F. Hasler, Dec 31 2014
(Haskell)
a252477 n = a252477_list !! (n-1)
a252477_list = map (floor . recip) $ zipWith (-) (tail rs) rs
where rs = map (sqrt . fromIntegral) a000040_list
-- Reinhard Zumkeller, Jan 04 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Dec 31 2014
STATUS
approved