Prime Factor and Exponent
For a positive integer $n \gt 1$, let $p(n)$ be the smallest prime dividing $n$, and let $\alpha(n)$ be its $p$-adic order, i.e. the largest integer such...
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John Rhodes
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August 16, 2023
Freshman’s Product
If $a,b$ are two nonnegative integers with decimal representations $a=(\dots a_2a_1a_0)$ and $b=(\dots b_2b_1b_0)$ respectively, then the freshman’s product of $a$ and $b$, denoted $a\boxtimes b$, is...
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John Rhodes
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August 16, 2023
Digit Sum Division
For a positive integer $n$, $d(n)$ is defined to be the sum of the digits of $n$. For example, $d(12345)=15$. Let $\displaystyle F(N)=\sum_{n=1}^N \frac n{d(n)}$. You are...
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John Rhodes
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August 16, 2023
Saving Paper
When wrapping several cubes in paper, it is more efficient to wrap them all together than to wrap each one individually. For example, with 10 cubes of...
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John Rhodes
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August 16, 2023
Conjunctive Sequences
Let ‘$\&$’ denote the bitwise AND operation. For example, $10\,\&\, 12 = 1010_2\,\&\, 1100_2 = 1000_2 = 8$. We shall call a finite sequence of non-negative integers...
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John Rhodes
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August 16, 2023
Ruff Numbers
Let $S_k$ be the set containing $2$ and $5$ and the first $k$ primes that end in $7$. For example, $S_3 = \{2,5,7,17,37\}$. Define a $k$-Ruff number...
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John Rhodes
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August 16, 2023
Balanceable $k$-bounded Partitions
A $k$-bounded partition of a positive integer $N$ is a way of writing $N$ as a sum of positive integers not exceeding $k$. A balanceable partition is...
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John Rhodes
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August 16, 2023
Pseudo Geometric Sequences
We define a pseudo-geometric sequence to be a finite sequence $a_0, a_1, \dotsc, a_n$ of positive integers, satisfying the following conditions: $n \geq 4$, i.e. the sequence...