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%I #25 May 18 2020 04:32:32
%S 1,1,1,2,1,2,1,2,2,4,2,2,3,2,4,4,1,2,3,3,4,4,2,4,4,4,4,4,4,4,5,5,6,4,
%T 5,7,3,6,6,8,5,5,7,4,6,7,6,7,6,6,5,9,7,7,6,7,7,6,8,8,7,7,8,9,11,7,8,
%U 10,8,11,8,7,7,10,11,12,4,9,11,6,9,9,10,8,9,8,11,8,8,9,10,8,13,10,9,10,14,12
%N Number of primes between n^2 and (n+1/2)^2.
%C For small values of n, these numbers exhibit higher and lower values as n increases. Conjectures: After n=17 a(n) > 1. There exists an n_1 such that a(n) is < a(n+1) for all n >= n_1.
%C Same as the number of primes between n^2 and n^2+n. Oppermann conjectured in 1882 that a(n)>0. - _T. D. Noe_, Sep 16 2008
%D Paulo Ribenboim, The New Book of Prime Number Records, 3rd ed., 1995, Springer, p. 248.
%H T. D. Noe, <a href="/A089610/b089610.txt">Table of n, a(n) for n=1..10000</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Oppermann%27s_conjecture">Oppermann's conjecture</a>
%t a[n_] := PrimePi[(n + 1/2)^2] - PrimePi[n^2]; Table[ a@n, {n, 100}] (* _Robert G. Wilson v_, May 04 2009 *)
%o (PARI) a(n) = primepi(n^2+n) - primepi(n^2); \\ _Michel Marcus_, May 18 2020
%o (Haskell)
%o a089610 n = sum $ map a010051' [n^2 .. n*(n+1)]
%o -- _Reinhard Zumkeller_, Jun 07 2015
%Y Cf. A010051, A014085, A094189, A108309.
%K easy,nonn
%O 1,4
%A _Cino Hilliard_, Dec 30 2003
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