A089610 Number of primes between n^2 and (n+1/2)^2.

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, 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, 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

Offset: 1

Author

Cino Hilliard, Dec 30 2003

Keywords

• easy
• nonn

Comments

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.

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

References

• Paulo Ribenboim, The New Book of Prime Number Records, 3rd ed., 1995, Springer, p. 248.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 183.

Links

• T. D. Noe, Table of n, a(n) for n=1..10000
• Wikipedia, Oppermann's conjecture

Crossrefs

Cf. A010051, A014085, A094189, A108309.

Programs

Haskell

a089610 n = sum $ map a010051' [n^2 .. n*(n+1)]
-- Reinhard Zumkeller, Jun 07 2015

Mathematica

a[n_] := PrimePi[(n + 1/2)^2] - PrimePi[n^2]; Table[ a@n, {n, 100}] (* Robert G. Wilson v, May 04 2009 *)

PARI

a(n) = primepi(n^2+n) - primepi(n^2); \\ Michel Marcus, May 18 2020