Drone Delivery
A depot uses $n$ drones to disperse packages containing essential supplies along a long straight road. Initially all drones are stationary, loaded with a supply package. Every...
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John Rhodes
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August 16, 2023
Slowly Converging Series
For a non-negative integer $k$, define \[ E_k(q) = \sum\limits_{n = 1}^\infty \sigma_k(n)q^n \] where $\sigma_k(n) = \sum_{d \mid n} d^k$ is the sum of the $k$-th...
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John Rhodes
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August 16, 2023
High Powers of Irrational Numbers
Given is the function $f(a,n)=\lfloor (\lceil \sqrt a \rceil + \sqrt a)^n \rfloor$. $\lfloor \cdot \rfloor$ denotes the floor function and $\lceil \cdot \rceil$ denotes the ceiling...
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John Rhodes
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August 16, 2023
Unpredictable Permutations
Consider all permutations of $\{1, 2, \ldots N\}$, listed in lexicographic order.For example, for $N=4$, the list starts as follows: $$\displaylines{ (1, 2, 3, 4) \\ (1,...
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John Rhodes
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August 16, 2023
Number Splitting
We define an $S$-number to be a natural number, $n$, that is a perfect square and its square root can be obtained by splitting the decimal representation...
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John Rhodes
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August 16, 2023
Unreachable Numbers
Consider the equation $17^pa+19^pb+23^pc = n$ where $a$, $b$, $c$ and $p$ are positive integers, i.e. $a,b,c,p \gt 0$. For a given $p$ there are some values...
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John Rhodes
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August 16, 2023
Summation of a Modular Formula
For an odd prime $p$, define $f(p) = \left\lfloor\frac{2^{(2^p)}}{p}\right\rfloor\bmod{2^p}$ For example, when $p=3$, $\lfloor 2^8/3\rfloor = 85 \equiv 5 \pmod 8$ and so $f(3) = 5$. Further...
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John Rhodes
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August 16, 2023
Sextuplet Norms
Let $f(n)$ be the number of $6$-tuples $(x_1,x_2,x_3,x_4,x_5,x_6)$ such that: All $x_i$ are integers with $0 \leq x_i < n$ $\gcd(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2,\ n^2)=1$ Let $\displaystyle G(n)=\displaystyle\sum_{k=1}^n \frac{f(k)}{k^2\varphi(k)}$ where...