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A074976 a(n) = round(1/(sqrt(prime(n+1)) - sqrt(prime(n)))).

%I A074976 #21 Feb 16 2025 08:32:47
%S A074976 3,2,2,1,3,2,4,2,2,5,2,3,6,3,2,2,8,3,4,8,3,4,3,2,5,10,5,10,5,2,6,4,12,
%T A074976 2,12,4,4,6,4,4,13,3,14,7,14,2,2,7,15,8,5,15,3,5,5,5,16,6,8,17,3,2,9,
%U A074976 18,9,3,6,4,19,9,6,5,6,6,10,7,5,10,5,4,20,4,21,7,10,7,5,11,21,11,4,5,11,6,11
%N A074976 a(n) = round(1/(sqrt(prime(n+1)) - sqrt(prime(n)))).
%C A074976 If Andrica's conjecture is true, each term in the sequence is >=1.
%C A074976 Since Andrica's conjecture is bounded below only by zero, a(n) is not bounded above.
%D A074976 C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 482.
%H A074976 Reinhard Zumkeller, <a href="/A074976/b074976.txt">Table of n, a(n) for n = 1..10000</a>
%H A074976 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AndricasConjecture.html">Andrica's conjecture</a>
%F A074976 Conjecture: for n>=4, a(n)>=2. More generally, for any m >=1, the set of k such that a(k)=m is finite. I.e., if n>=217, a(n)>=3; if n>=263, a(n)>=4; if n>=590, a(n)>=5; if n>=3385, a(n)>=6; ...
%e A074976 a(1) = round(1/(sqrt(3) - sqrt(2))) = round(1/(1.7320.. - 1.4142..)) = round(1/0.3178..) = round(3.1462..) = 3.
%t A074976 Round[1/(Sqrt[#[[2]]]-Sqrt[#[[1]]])]&/@Partition[Prime[Range[100]],2,1] (* _Harvey P. Dale_, May 30 2022 *)
%o A074976 (PARI) a(n) = round(1/(sqrt(prime(n+1))-sqrt(prime(n)))) \\ _Michel Marcus_, May 22 2013
%o A074976 (Haskell)
%o A074976 a074976 n = a074976_list !! (n-1)
%o A074976 a074976_list = map (round . recip) $ zipWith (-) (tail rs) rs
%o A074976                where rs = map (sqrt . fromIntegral) a000040_list
%o A074976 -- _Reinhard Zumkeller_, Jan 04 2015
%Y A074976 Cf. A000040, A252477.
%K A074976 nonn
%O A074976 1,1
%A A074976 _Werner D. Sand_ and _Benoit Cloitre_, Oct 06 2002