bibkey: hilliard2003a089610 authors: Cino Hilliard year: 2003 title: “OEIS A089610, Number of primes between n^2 and (n+1/2)^2” doi: null url: https://oeis.org/A089610 claim: “Number of primes between n^2 and (n+1/2)^2. 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 Cino Hilliard, Dec 30 2003” strata_touched:

• D5/S3/PrimeGaps/HilliardSquareIntervalPrimeCountEventualIncreaseRefutation license: citation-only triage: anchor

OEIS A089610

The NAME of A089610 defines the square-interval prime count:

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

The COMMENTS lines give the conjectures and the equivalent interval endpoint:

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

The AUTHOR line identifies the contributor:

Cino Hilliard, Dec 30 2003

Only the second conjecture, eventual strict increase, is refuted: every window [M, 2M+2) contains a non-increase because a(2k) <= k; the first conjecture and Oppermann’s positivity conjecture are untouched.

Verified locator

• URL: https://oeis.org/A089610
• NAME (verbatim): Number of primes between n^2 and (n+1/2)^2.
• COMMENTS (verbatim): 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.
• COMMENTS (verbatim): 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
• AUTHOR (verbatim): Cino Hilliard, Dec 30 2003