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A089610 Number of primes between n^2 and (n+1/2)^2. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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.

LINKS

T. D. Noe, Table of n, a(n) for n=1..10000

Wikipedia, Oppermann's conjecture

MATHEMATICA

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

PROG

(PARI) pbetweensq2(n) = { for(x=1, n, c=0; for(y=floor(x)^2, (x+.5)^2, if(isprime(y), c++) ); print1(c", ") ) }

(Haskell)

a089610 n = sum $ map a010051' [n^2 .. n*(n+1)]

-- Reinhard Zumkeller, Jun 07 2015

CROSSREFS

Cf. A014085, A094189, A108309.

Cf. A010051.

Sequence in context: A230224 A206941 A143179 * A102566 A279371 A134156

Adjacent sequences:  A089607 A089608 A089609 * A089611 A089612 A089613

KEYWORD

easy,nonn

AUTHOR

Cino Hilliard, Dec 30 2003

STATUS

approved

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Last modified June 16 14:56 EDT 2019. Contains 324152 sequences. (Running on oeis4.)