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Metadata 
bibkey: oudra2023a248186
authors: Clark Kimberling; Ridouane Oudra
year: 2023
title: OEIS A248186, least reciprocal-product threshold index
doi: null
url: https://oeis.org/A248186/internal
claim: "Conjecture: a(n) = floor(r*n + r^2/n - 1), where r = 3^(-1/3), for n > 2. - _Ridouane Oudra_, Oct 06 2023"
strata_touched:
  - D5/S3/ArithSums/OudraA248186CubicFloor
license: citation-only
triage: anchor

Verified locator

Source: https://oeis.org/A248186/internal

OEIS A248186 revision 18, NAME (%N) and FORMULA (%F), unpaginated. The entry is by Clark Kimberling; the formula conjecture is credited to Ridouane Oudra.

Source statement

Least k such that 1/18 - Sum_{h=1..k} 1/(h*(h+1)(h+2)(h+3)) < 1/n^3.

Conjecture: a(n) = floor(r*n + r^2/n - 1), where r = 3^(-1/3), for n > 2. - Ridouane Oudra, Oct 06 2023

The offset is 1,5. The source program selects from Mathematica’s Range[z], so the least index is positive. Its data start with a(1)=a(2)=1. The residual is a real finite sum; integer floor is taken after real subtraction. The assertion includes existence of the sequence at every positive index.

Mathematical mechanism

Finite telescoping gives the residual 1/(3*(k+1)*(k+2)*(k+3)). With r=3^(-1/3) and b=r*n+r^2/n, expansion gives 3*(b^3-b)=n^3+1/(3*n^3). For n>2 the excess is strictly between zero and one. Monotonicity of 3*(x^3-x) for x>=1, together with integrality at integer arguments, brackets the threshold by the two consecutive integers floor(b) and floor(b)+1. The least admissible index is floor(b)-1.

The primary source is this OEIS FORMULA conjecture. Chai Wah Wu’s “Algorithms for complementary sequences,” arXiv:2409.05844v10, https://arxiv.org/abs/2409.05844v10, and its journal version, Integers 25 (2025), A95, DOI 10.5281/zenodo.17535229, contain no A248186 or Oudra statement in the source texts checked in issue #24012. They are comparison literature. The earlier A056556 formula concerns integer arguments and does not by direct substitution settle this rounding formula. Reciprocal-product telescoping is classical background; the claimed settlement concerns the exact cubic floor.

The bounded literature screen is recorded in https://github.com/the-omega-institute/trureturing/issues/24012. Global completeness, blocked sources and unindexed work are ASSUMED-UNVERIFIED. No conclusion about A248187 or A248188 is asserted here.

Supporting computation

Repository: the-omega-institute/trureturing-experiments. Commit: 4093bd7a859a5be8bec8c9615ef3150c9e427071. Program: docs/reports/oudra-a248186/check.py. SHA-256: eda2a4d6a3054c01f38b36fd5c1fe4b4e058ef0a7a94f72f2b45ce5c7364e4d1. The preregistration reports Python exit code 0 and final output ALL_CHECKS_OK, including all 9,998 cases 3<=n<=10000. These finite computations support the argument and do not prove its universal conclusion.