KG

Skilled in Python, number theory, code golf, and messing about with calculators.

less than 1 minute read

Marvin is interested in numbers whose sum of their digits equals the product of their digits. Call a nonnegative integer a $b$-Marvin number if it has this property when written in base $b$. Marvin noticed already that all single digits in base $b$ are trivially $b$-Marvin numbers.

  1. Prove there are no nontrivial (i.e. more than one digit) $b$-Marvin numbers when $b \leq 2$.
  2. Prove there are infinitely many $b$-Marvin numbers when $b \geq 3$.
  3. Prove there are only finitely many $b$-Marvin numbers not containing a 1 in any base.
  4. Prove that, for all positive integers $n$, there exists a base $b$ for which $n$ is a $b$-Marvin number.

You May Also Enjoy

Teensy Problems: Sum of Factorials

1 minute read

Let $\mathrm{sf}(n)$ be the sum-of-factorials function: $0! + 1! + … n!$. Its first few values are $1$, $2$, $4$, $10$, $34$, ….

Teensy Problems: Falling Leaves

less than 1 minute read

Autumn and Yvonne love to play “falling leaves” this time of year. They play on a tree containing some number of leaves (possibly none) on its left and right...

Teensy Problems: Coprime Labeling

1 minute read

Let $G$ be a graph with $n$ vertices. We say a graph is coprimely labeled if its vertices are labeled using all the integers from 1 to $n$ in such a way wher...

Teensy Problems: Counting Carol

less than 1 minute read

Carol really likes counting, but can only keep track of so many numbers at once. She’s devised a family of sequences that matches these aptitudes, in the fol...