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%I #10 Jan 04 2015 00:18:25
%S 3,1,2,1,3,1,4,2,1,5,1,3,6,3,2,2,7,2,4,8,2,4,3,2,4,10,5,10,5,1,5,3,11,
%T 2,12,4,4,6,4,4,13,2,13,6,14,2,2,7,15,7,5,15,3,5,5,5,16,5,8,16,3,2,8,
%U 17,8,2,6,3,18,9,6,4,6,6,9,6,4,9,5,4,20,4,20,6,10,7,5,10,21,10,3,5
%N Integer part of 1/(sqrt(prime(n+1))-sqrt(prime(n))).
%C 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.
%C Sequence A074976 is based on the same idea (rounding to the nearest integer instead).
%C 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, ...
%C Equal to the integer part of (A000006(n+1)+A000006(n))/(prime(n+1)-prime(n)) for most indices; exceptions are 1, 129, 1667, 2004, 2088, 2334, 3377, 3585, 3695, 3834, 4978, 7057, 7950, 8103, 9525, 9805,...
%H M. F. Hasler, <a href="/A252477/b252477.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AndricasConjecture.html">Andrica's conjecture</a>
%e a(1) = floor(1/(sqrt(3) - sqrt(2))) = floor(1/(1.73-1.41)) = floor(1/0.32) = floor(3.15) = 3.
%e a(2) = floor(1/(sqrt(5) - sqrt(3))) = floor(1/(2.236-1.732)) = floor(1/0.504) = floor(1.98) = 1.
%o (PARI) a(n)=1\(sqrt(prime(n+1))-sqrt(prime(n))) \\ _M. F. Hasler_, Dec 31 2014
%o (Haskell)
%o a252477 n = a252477_list !! (n-1)
%o a252477_list = map (floor . recip) $ zipWith (-) (tail rs) rs
%o where rs = map (sqrt . fromIntegral) a000040_list
%o -- _Reinhard Zumkeller_, Jan 04 2015
%Y Cf. A000040, A074976.
%K nonn
%O 1,1
%A _M. F. Hasler_, Dec 31 2014
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