Computational Mathematics

John Rhodes
August 15, 2023
Computational Mathematics

Double Pandigital Number Divisible by $11$

We call a positive integer double pandigital if it uses all the digits $0$ to $9$ exactly twice (with no leading zero). For example, $40561817703823564929$ is one...

John Rhodes
August 15, 2023
Computational Mathematics

Jumping Frog

There are $n$ stones in a pond, numbered $1$ to $n$. Consecutive stones are spaced one unit apart. A frog sits on stone $1$. He wishes to...

John Rhodes
August 15, 2023
Computational Mathematics

Common Factors Between Two Sequences

Let $G(a, b)$ be the smallest non-negative integer $n$ for which $\operatorname{\mathbf{gcd}}$Greatest common divisor$(n^3 + b, (n + a)^3 + b)$ is maximized. For example, $G(1, 1)...

John Rhodes
August 15, 2023
Computational Mathematics

Unbalanced Nim

Alice and Bob have enjoyed playing Nim every day. However, they finally got bored of playing ordinary three-heap Nim. So, they added an extra rule: – Must...

John Rhodes
August 15, 2023
Computational Mathematics

Sums of Power Sums

Let $f_k(n)$ be the sum of the $k$th powers of the first $n$ positive integers. For example, $f_2(10) = 1^2 + 2^2 + 3^2 + 4^2 +...

John Rhodes
August 15, 2023
Computational Mathematics

Maximum Number of Divisors

Let $d(n)$ be the number of divisors of $n$. Let $M(n,k)$ be the maximum value of $d(j)$ for $n \le j \le n+k-1$. Let $S(u,k)$ be the...

John Rhodes
August 15, 2023
Computational Mathematics

Arithmetic Derivative

The arithmetic derivative is defined by $p^\prime = 1$ for any prime $p$ $(ab)^\prime = a^\prime b + ab^\prime$ for all integers $a, b$ (Leibniz rule) For...

John Rhodes
August 15, 2023
Computational Mathematics

Repeated Permutation

We define a permutation as an operation that rearranges the order of the elements $\{1, 2, 3, …, n\}$. There are $n!$ such permutations, one of which...