bibkey: noe2013a222759
authors: T. D. Noe
year: 2013
title: "OEIS A222759: first growth assertion for literal binomial remainders"
doi: null
url: https://oeis.org/A222759
claim: "It appears that, for k > 2 and n >= prime(prime(k)^3), then a(n) >= k."
strata_touched:
  - D5/S3/Arith/PrimeRemainders/OeisA222759Growth
license: citation-only
triage: anchor

The first growth assertion of A222759

The NAME is:

Conjectured number of primes p for which binomial(n*p,p) (mod p^3) does not equal n.

The first COMMENT is:

It appears that, for k > 2 and n >= prime(prime(k)^3), then a(n) >= k.

These quotations refer to revision 12, November 2, 2024, of the entry by T. D. Noe. The offset is 1,2. The source is https://oeis.org/search?q=id%3AA222759&fmt=text .

Prime(j) is the one-based j-th prime, with Prime(1)=2. The parentheses in the threshold mean P(P(k)^3), not P(P(k))^3. The related threshold sequence A096328 defines Prime(p^3) for p=prime(n), confirming this placement.

The Mathematica predicate is Mod[Binomial[n*#,#], #^3] != n. For a positive modulus m, Mod returns an integer from zero to m-1. Thus this predicate compares the literal remainder with n itself. The finite prime range in the sample code is not a universal quantifier. Operator references are https://reference.wolfram.com/language/ref/Mod.html and https://reference.wolfram.com/language/ref/Prime.html .

The exact mathematical event is: for every integer k>=3 and every integer n>=1 with n>=P(P(k)^3), there exist k pairwise distinct positive primes q whose binomial(n*q,q) remainder modulo q^3 differs from the integer n. The formal result uses an injection from Fin k into the natural numbers, with this predicate at every index. This directly expresses at least k witnesses, without using the natural-valued cardinality of an infinite set or adding a finiteness premise. Positive integer inputs correspond exactly to the natural inputs with the displayed lower bounds.

The source scope is only this first growth assertion. Other comments, exact term values, the sample table, and A222760 are separate assertions.

Verified locator

Source: https://oeis.org/A222759

OEIS A222759, revision 12, November 2, 2024, first COMMENT only. The quoted NAME fixes the literal remainder predicate, and the first COMMENT fixes the nested prime threshold and universal growth assertion.