Related papers: On Two OEIS Conjectures

In this note, we prove a conjecture proposed by Tao Zhang, Shuxing Li, Tao Feng and Gennian Ge, IEEE Transaction on Information Theory, vol. 60, no. 5, May 2014. This conjecture is about the cross correlation distribution of ternary…

We prove a recent conjecture of Sean A. Irvine about a nonlinear recurrence, using mechanized guessing and verification. The theorem-prover Walnut plays a large role in the proof.

For any integer $r \geq 1$, the sequence of numbers $\{{c^{(r)}_{k}}\}_{k \geq 0} $ is defined implicitly by [\sum_k\binom{n}{k}^r\binom{n+k}{k}^r = \sum_k\binom{n}{k}\binom{n+k}{k}c^{(r)}_k,\quad n=0,1,2,...] Asmus Schmidt conjectured that…

This is a detailed survey on the QWEP conjecture and Connes' embedding problem. Most of contents are taken from Kirchberg's paper [Invent. Math. 112 (1993)].

Let $n$ be a positive integer and let $S$ be a sequence of $n$ integers in the interval $[0,n-1]$. If there is an $r$ such that any nonempty subsequence with sum $\equiv 0$ $\pmod n$ has length $=r,$ then $S$ has at most two distinct…

We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…

In this paper, we give recurrence relations and identities for some integer sequences related to Ward numbers such as Ward-Lah numbers, varied Ward numbers and binomial Ward numbers. Most of the sequences are entered in the On-Line…

This article is a concise review of covariant string field theory prepared for the Encyclopedia of Mathematical Physics, Elsevier (2006). Referencing follows the publisher's guidelines.

For Hilbert, the consistency of a formal theory T is an infinite series of statements "D is free of contradictions" for each derivation D and a consistency proof is i) an operation that, given D, yields a proof that D is free of…

In this paper we present some observations about the well-known Goldbach conjecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also…

Hypertrees are linear hypergraphs where every two vertices are connected by a unique path. Elliott and R\"{o}dl conjectured that for any given $\mu>0$, there exists $n_0$ such that the following holds. Every $n$-vertex Steiner triple system…

This is a collection of variants of Schanuel's conjecture and the known dependencies between them. It was originally written in 2007, and made available for a time on my webpage. I have been asked by a few people to make it available again…

This paper gives a brief description of the author's database of integer sequences, now over 35 years old, together with a selection of a few of the most interesting sequences in the table. Many unsolved problems are mentioned.

Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of $0021$-avoiding inversion sequences can be enumerated by the OEIS entry…

In this paper, we explore two conjectures about Rademacher sequences. Let $(\epsilon_i)$ be a Rademacher sequence, i.e., a sequence of independent $\{-1,1\}$-valued symmetric random variables. Set $S_n=a_1\epsilon_1+\cdots+a_n\epsilon_n$…

We relate binary words with a given number of subsequences to continued fractions of rational numbers with a given denominator. We deduce that there are binary strings of length $O(\log n \log \log n)$ with exactly $n$ subsequences; this…

We settle 22 conjectures of Cohen about cyclic numbers (positive integers $n$ with $\gcd(n,\varphi(n))=1$), proving 16 and disproving 6, and we completely resolve a related OEIS problem about sequences whose running averages are Fibonacci…

Based on large N Chern-Simons/topological string duality, in a series of papers, J.M.F. Labastida, M. Marino, H. Ooguri and C. Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite…

Seymour's distance two conjecture states that in any digraph there exists a vertex (a "Seymour vertex") that has at least as many neighbors at distance two as it does at distance one. We explore the validity of probabilistic statements…

Assuming the existence of Siegel zeros, we prove that there exists an increasing sequence of positive integers for which Chowla's Conjecture on $k$-point correlations of the Liouville function holds. This extends work of Germ\'an and…