This note considers possible arrangements of the sectors of a generalised dartboard. The sum of the pth powers of the absolute differences of the numbers on adjacent sectors is introduced as a penalty cost function and a ...

A number n > 1 is harmonic if sigma(n) vertical bar n tau(n), where tau(n) and sigma(n) are the number of positive divisors of n and their sum, respectively. It is known that there are no odd harmonic numbers up to 10(16). ...

Let tau(n) denote the number of positive divisors of a natural number n > 1 and let sigma( n) denote their sum. Then n is superharmonic if sigma(n) vertical bar n(k)tau(n) for some positive integer k. We deduce numerous ...