Tricoloured Coin Fountains
An arrangement of coins in one or more rows with the bottom row being a block without gaps and every coin in a higher row touching exactly...
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John Rhodes
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August 15, 2023
Prime Triples and Geometric Sequences
Let $S(n) = \sum a + b + c$ over all triples $(a, b, c)$ such that: $a$, $b$ and $c$ are prime numbers. $a \lt b...
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John Rhodes
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August 15, 2023
A Real Recursion
For every real number $a \gt 1$ is given the sequence $g_a$ by: $g_{a}(x)=1$ for $x \lt a$ $g_{a}(x)=g_{a}(x-1)+g_a(x-a)$ for $x \ge a$ $G(n)=g_{\sqrt {n}}(n)$ $G(90)=7564511$. Find...
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John Rhodes
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August 15, 2023
$5$-smooth Totients
$5$-smooth numbers are numbers whose largest prime factor doesn’t exceed $5$. $5$-smooth numbers are also called Hamming numbers. Let $S(L)$ be the sum of the numbers $n$...
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John Rhodes
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August 15, 2023
Dissonant Numbers
Let $d(p, n, 0)$ be the multiplicative inverse of $n$ modulo prime $p$, defined as $n \times d(p, n, 0) = 1 \bmod p$. Let $d(p, n,...
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John Rhodes
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August 15, 2023
Geoboard Shapes
A geoboard (of order $N$) is a square board with equally-spaced pins protruding from the surface, representing an integer point lattice for coordinates $0 \le x, y...
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John Rhodes
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August 15, 2023
Integral Median
$ABC$ is an integral sided triangle with sides $a \le b \le c$. $m_C$ is the median connecting $C$ and the midpoint of $AB$. $F(n)$ is the...
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John Rhodes
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August 15, 2023
Sums of Totients of Powers
Let $\varphi(n)$ be Euler’s totient function. Let $f(n)=(\sum_{i=1}^{n}\varphi(n^i)) \bmod (n+1)$. Let $g(n)=\sum_{i=1}^{n} f(i)$. $g(100)=2007$. Find $g(5 \times 10^8)$. To solve this problem, we can use dynamic programming...