Coin Partitions
Let $p(n)$ represent the number of different ways in which $n$ coins can be separated into piles. For example, five coins can be separated into piles in...
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John Rhodes
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August 15, 2023
Prime Summations
It is possible to write ten as the sum of primes in exactly five different ways: \begin{align} &7 + 3\\ &5 + 5\\ &5 + 3 +...
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John Rhodes
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August 15, 2023
Counting Summations
It is possible to write five as a sum in exactly six different ways: \begin{align} &4 + 1\\ &3 + 2\\ &3 + 1 + 1\\ &2...
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John Rhodes
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August 15, 2023
Digit Factorial Chains
The number $145$ is well known for the property that the sum of the factorial of its digits is equal to $145$: $$1! + 4! + 5!...
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John Rhodes
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August 15, 2023
Counting Fractions in a Range
Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n, d)=1$, it is called a reduced proper fraction....
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John Rhodes
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August 15, 2023
Counting Fractions
Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction. If...
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John Rhodes
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August 15, 2023
Ordered Fractions
Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction. If...
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John Rhodes
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August 15, 2023
Totient Permutation
Euler’s totient function, $\phi(n)$ [sometimes called the phi function], is used to determine the number of positive numbers less than or equal to $n$ which are relatively...