The OEIS A089610 Eventual Strict-Increase Conjecture
Abstract
Odd-prime counting rules out eventual strict increase for OEIS A089610.
Definition 1.1 (The square-interval prime count).
Formalization. D5/S3/PrimeGaps/HilliardSquareIntervalPrimeCountEventualIncreaseRefutation.a (✓ std3).
Citation. Cino Hilliard (2003). OEIS A089610, Number of primes between n^2 and (n+1/2)^2. URL: https://oeis.org/A089610.
Commentary.
The value a(n) counts the primes in the half-open interval (n^2, n^2+n]. Since (n+1/2)^2 = n^2+n+1/4, this interval contains the same integer primes as the printed OEIS interval.
Definition 1.2 (The eventual strict-increase conjecture).
Formalization. D5/S3/PrimeGaps/HilliardSquareIntervalPrimeCountEventualIncreaseRefutation.claim (✓ std3).
Citation. Cino Hilliard (2003). OEIS A089610, Number of primes between n^2 and (n+1/2)^2. URL: https://oeis.org/A089610.
Commentary.
The claim asks for a natural threshold N after which every successive value of a is strictly larger.
Theorem 1.3 (Eventual strict increase is impossible).
Proof. Machine-checked in Lean as D5/S3/PrimeGaps/HilliardSquareIntervalPrimeCountEventualIncreaseRefutation.result (✓ std3). ∎
Resolves. Problems/oeis-a089610-square-interval-prime-count-eventual-increase-refutation (refuted) by D5/S3/PrimeGaps/HilliardSquareIntervalPrimeCountEventualIncreaseRefutation.result.
Source. Repository-derived.
Acknowledgement. Cino Hilliard (2003). OEIS A089610, Number of primes between n^2 and (n+1/2)^2. URL: https://oeis.org/A089610.
Commentary.
At every even index, the odd primes in the defining interval inject into k positions, giving a(2k) <= k. Strict increase through a window from M to 2M+2 would instead force the final value above this bound. The first conjecture a(n) > 1 after n = 17 and Oppermann’s positivity conjecture are untouched.