Nearest integer to $n/\sqrt{2}$

formal-conjectures · oeis · ams-11

Nearest integer to $n/\sqrt{2}$, defined by $\lfloor n/\sqrt{2} + 1/2 \rfloor$.

Let $s(n) = \zeta(3) - \sum_{k=1}^n \frac{1}{k^3}$.
Conjecture: for $n \ge 1$, $s(a(n)) < \frac{1}{n^2} < s(a(n)-1)$, and the difference sequence of
A049473 consists solely of $0$'s and $1$'s, in positions given by the nonhomogeneous Beatty
sequences A001954 and A001953, respectively.
- Clark Kimberling, Oct 05 2014

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A049473
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/49473.lean
Local target: Corpus.OEIS49473.conjecture
Source SHA-256: d45c46663792eb38950707899080704d3d69bf8457e8ef9c8a2e1de85dd65844
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

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