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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 …

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 …

$n = 2673$ 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 …

$7 \sim 12$ via $3, 4$ $12 \sim 35$ via $5, 7$ $35 \sim 264$ via $11, 24$ $264 \sim 41$ via $8, 33$ $41 \sim 420$ via $20, 21$ $420 \sim 43$ via $15, 28$ $43 \sim 156$ via $4, 39$ $156 \sim 25$ via $ …

To be unambiguous about the order of multiplication, let $B(n) = A_1 A_2 \cdots A_n$. We have the D-finite recurrences $B(n)_{r,1} = (\frac{n}{n-1})^k B(n-1)_{r,1} + n^k B(n-2)_{r,1}$ $B(n)_{r,2} = B …

Experimenting with a CAS suggests an induction. In order to handle the induction, we need to consider the forms of the numbers involved. $\frac{4^m-1}{3} = 1 + 2^2 + 2^4 + \cdots + 2^{2m-2}$ alternate …

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 $p(463, x)$.

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 …

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 graph $G_k$ with vertex set $$\{u_1, \ldots, u_k, v_1, \ldots, v_k\}$$ and edges $$\{(u_1, v_1), \ldots, (u_1, v_k)\} \cup \{(u_2, v_1), \ldots, (u_k, v_{k-1})\} \cup \{(u_2, v_k), \ldots …

For any odd $p$, $q$ (not necessarily prime) the values modulo $2$ follow a cycle of order 3.

This is only empirical observation, but I was requested to post it as an answer rather than merely a comment. Define $b(n) = \frac{\det(A_n)}{n! \, \sigma(\operatorname{lcm}(1,\ldots,n))}$ for $n \ge …

I'm going to use $\operatorname{msb}$ (for most significant bit) as an alias of $f$. Since $q_i$ is a permutation, the property that $q_i(n)<2^k$ iff $n < 2^k$ is equivalent to $\operatorname{msb}(q_i …