Fractional Sequences
For any positive integer $k$, a finite sequence $a_i$ of fractions $x_i/y_i$ is defined by: $a_1 = 1/k$ and $a_i = (x_{i – 1} + 1) /...
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John Rhodes
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August 15, 2023
The Totient of a Square Is a Cube
Consider the number $50$. $50^2 = 2500 = 2^2 \times 5^4$, so $\phi(2500) = 2 \times 4 \times 5^3 = 8 \times 5^3 = 2^3 \times 5^3$....
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John Rhodes
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August 15, 2023
Golomb’s Self-describing Sequence
The Golomb’s self-describing sequence $(G(n))$ is the only nondecreasing sequence of natural numbers such that $n$ appears exactly $G(n)$ times in the sequence. The values of $G(n)$...
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John Rhodes
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August 15, 2023
Crazy Function
For fixed integers $a, b, c$, define the crazy function $F(n)$ as follows: $F(n) = n – c$ for all $n \gt b$ $F(n) = F(a +...
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John Rhodes
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August 15, 2023
Peredur Fab Efrawg
“And he came towards a valley, through which ran a river; and the borders of the valley were wooded, and on each side of the river were...
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John Rhodes
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August 15, 2023
Cutting Rectangular Grid Paper
A rectangular sheet of grid paper with integer dimensions $w \times h$ is given. Its grid spacing is $1$. When we cut the sheet along the grid...
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John Rhodes
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August 15, 2023
Totient Stairstep Sequences
Let $\{a_1, a_2, \dots, a_n\}$ be an integer sequence of length $n$ such that: $a_1 = 6$ for all $1 \le i \lt n$: $\phi(a_i) \lt \phi(a_{i...
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John Rhodes
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August 15, 2023
Maximix Arrangements
A train is used to transport four carriages in the order: ABCD. However, sometimes when the train arrives to collect the carriages they are not in the...