bibkey: browne2016a164660
authors: Anthony Browne; Wolfdieter Lang
year: 2016
title: "OEIS A164660: Chebyshev row numerators and Browne's floor product"
doi: null
url: https://oeis.org/search?q=id:A164660&fmt=text
claim: "a(n+1) = Product_{k=1..n} ( 1-2*(floor(k^n/n)-floor((k^n -1)/n)) ) = (-1)^(A003557(n)) for n>0 (conjecture)."
strata_touched:
  - D5/S3/ArithSums/BrowneA164660ChebyshevFloorProduct
license: citation-only
triage: anchor

Browne’s Chebyshev numerator and floor product

Verified locator

https://oeis.org/search?q=id:A164660&fmt=text

OEIS A164660, revision 19, supplies these verbatim FORMULA lines:

a(n) = numerator(Sum_{m=1..n+1} IT(n,m)), n>=0, with IT(n,m):= A164658(n,m)/A164659(n,m) (coefficient triangle from the indefinite integral Integral_{x} T(n,x), n>=0, in lowest terms).

a(n+1) = Product_{k=1..n} ( 1-2*(floor(k^n/n)-floor((k^n -1)/n)) ) = (-1)^(A003557(n)) for n>0 (conjecture). - Anthony Browne, May 29 2016

The record has no page numbers. Wolfdieter Lang introduced the sequence; Anthony Browne authored the final conjectural FORMULA.

The coefficient definition is given in A164658, revision 21:

a(n,m) = numerator(b(n,m)), with int(T(n,x))= sum(b(n,m)*x^m,m=1..n+1), n>=0, where T(n,x) are Chebyshevs polynomials of the first kind.

A164659, revision 10, gives the denominator of the same coefficients. The primitive has constant coefficient zero. Thus its column j is the coefficient of degree j-1 in T(m,x), divided by j, for 1 <= j <= m+1. The row sum is rational, and its numerator is the reduced signed numerator with positive denominator. The floor product uses rational subtraction and division before taking integer floors. Its index set includes both endpoints.

A003557, revision 172, identifies its term as follows:

n divided by largest squarefree divisor of n; if n = Product p(k)^e(k) then a(n) = Product p(k)^(e(k)-1), with a(1) = 1.

a(n) = n/rad(n) = n/A007947(n) = sqrt(J_2(n)/J_2(rad(n))), where J_2(n) is A007434. - Enrique Pérez Herrero, Aug 31 2010

The Lean exponent is natural division by the existing natural radical, including radical(1)=1. The theorem asserts both source equalities for every positive natural index, including one. It derives the numerator through a polynomial primitive and the product through counting multiples of the radical; divisibility by four identifies their common sign.

Attribution and license

The quotations are attributed to Anthony Browne, Wolfdieter Lang, Enrique Pérez Herrero and the OEIS Foundation Inc., under CC BY-SA 4.0. The source supplies the definitions and conjecture, not the repository proof.

Scope

The source is an OEIS record, with no identified source arXiv paper or DOI. The literature screen in issue #21736 found no complete settlement within its stated scope. Completeness is ASSUMED-UNVERIFIED: citation enumeration failed at OpenAlex and Semantic Scholar; non-arXiv later-author works and journal versions were not comprehensively enumerated. No originality or worldwide absence claim follows from these searches.