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$n = 2673$ has largest prime factor $11$ whose cube is $1331$. $n^2 - 1 = 7144928$ has largest prime factor $191$ whose cube is $6967871$. $n^2 + 1 = 7144930$ has largest prime factor $61$ whose cube …

$$\sum_{k=0}^{N-1} 3^k \, 2^{-L_{k+1}} \equiv 1 \pmod{3^N}$$ is equivalent to a series of equations modulo powers of $3$: $$\begin{eqnarray*} 2^{-L_1} &\equiv& 1 \pmod{3} \\ 3 \cdot 2^{-L_2} &\equiv& …

Hint: it's always worth checking the Online Encyclopedia of Integer Sequences. For $r=3$ the values of $N$ are the sequence A002426. There's a wealth of literature references, a number of comments whi …

Split $x_i$ into $z$ zeroes and a partition of $n$ into $k$ (non-zero) parts, $\lambda_j$. Then your equality can be rewritten as $$z(z-1) + kz + \mathop{\sum\sum}_{i \neq j} \lambda_i^{\lambda_j} = n …

Gauss' contiguous relations provide a basis for finding a linear relationship between three functions of the form ${}_2F_1(a+k, b+l, c+m, z)$, henceforth $\mathbf{F}\left(\begin{matrix}a+k, b+l \\ c+m …

Consider the case where $(0,0)$ and $(0,1)$ are winning (i.e. P) positions. There's a biphasic structure. Let $\operatorname{off}_w(b) = 3 \lfloor \frac b2 \rfloor + (b \bmod 1)$. Then if $w < \operat …

Original question Is this sequence strictly increasing? No. n difference smaller half 16 16753029012720 [3, 5, 6, 7, 9, 10, 11, 13, 15, 18, 19, 21, 25, 27, 29, 30] 17 10176199188480 [4, 6, 7, 8 …

A brief comment on notation: "can be averaged by" is a reflexive partial order. I shall use the notation $a \succeq b$ to indicate that $a$ can be averaged by $b$, and save $\geq$ for the usual total …

Counterexample: $463 \in b(n)$ (it's a prime and $464 = 2^4 \cdot 29$ is not squarefree), but $463 \not \in a(n)$ because it's a factor of the GCD of the coefficients of $p(463, x)$.

If $s(n)$ is the largest integer such that $s(n)^2 \mid \binom{2n}{n}$, then the main result of Sárközy, A. (1985). On divisors of binomial coefficients, I. Journal of Number Theory, 20(1), 70-80 is …

By identifying the lattice points with numbers of the form $x - y\omega$, $\omega = e^{2\pi i / 3}$, $x, y \in \mathbb{Z}$, we find that we want to count Diophantine solutions to $x^2 + xy + y^2 \le r …

The map $s(r) = \frac{n+2 + nr - r^2}{1 + (n+2)r}$ cyclically permutes the roots. This map is given in [2], and I found it through the reference in [1]. It turns out to give the correct order. Explici …

The review Bounds on Factors of $\mathbb{Z}[x]$ by John Abbott gives various such bounds and shows that none of them is strictly better than the others.

The $n_i$ are any sequence of distinct integers, indexed by $i$. The point is that if $R = r^{kl}$ then $R$ is a power of $p$, so any sum of distinct powers of $R$ is a 0-1 number in base $p$; and sym …

Suppose this is known for all Egyptian fractions with minimal representation in $k$ fractions. Then if there's a counterexample in $k+1$ fractions $\frac{1}{n_1} + \frac{1}{n_2} + \cdots \frac{1}{n_{k …