Numbers of the Form $a^2b^3$
Define $F(n)$ to be the number of integers $x≤n$ that can be written in the form $x=a^2b^3$, where $a$ and $b$ are integers not necessarily different and...
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John Rhodes
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August 16, 2023
Square Prime Factors II
For an integer $n$, we define the square prime factors of $n$ to be the primes whose square divides $n$. For example, the square prime factors of...
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John Rhodes
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August 16, 2023
Square Prime Factors
For an integer $n$, we define the square prime factors of $n$ to be the primes whose square divides $n$. For example, the square prime factors of...
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John Rhodes
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August 16, 2023
Constrained Permutations
Let $(p_1 p_2 \ldots p_k)$ denote the permutation of the set ${1, …, k}$ that maps $p_i\mapsto i$. Define the length of the permutation to be $k$;...
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John Rhodes
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August 16, 2023
Crossed Lines
Given a set, $L$, of unique lines, let $M(L)$ be the number of lines in the set and let $S(L)$ be the sum over every line of...
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John Rhodes
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August 16, 2023
Scatterstone Nim
Alice and Bob are playing a modified game of Nim called Scatterstone Nim, with Alice going first, alternating turns with Bob. The game begins with an arbitrary...
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John Rhodes
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August 16, 2023
Open Chess Positions
A position in chess is an (orientated) arrangement of chess pieces placed on a chessboard of given size. In the following, we consider all positions in which...
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John Rhodes
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August 16, 2023
Counting Products
Consider the set $S$ of all possible products of $n$ positive integers not exceeding $m$, that is $S=\{ x_1x_2\cdots x_n \mid 1 \le x_1, x_2, \dots, x_n...