pöördvandefunktsioon

Pöördvandefunktsioon, also known as the inverse sum-of-divisors function, is a mathematical function that for a given positive integer n, finds all positive integers m such that the sum of the divisors of m is equal to n. The sum of divisors function, denoted by sigma or $\sigma(m)$, is the sum of all positive divisors of m, including 1 and m itself. Therefore, the pöördvandefunktsioon seeks to solve the equation $\sigma(m) = n$ for m.

This function is not well-behaved in the same way as many other number-theoretic functions. For a given integer n, there might be no solutions for m, a unique solution, or multiple solutions. For example, if n = 12, the equation $\sigma(m) = 12$ has two solutions: m = 6 (divisors are 1, 2, 3, 6; sum is 12) and m = 11 (divisors are 1, 11; sum is 12). If n = 1, there are no solutions since the smallest possible sum of divisors for any m > 1 is 1+2 = 3 (for m=2). If n = 3, the unique solution is m = 2.

The study of the pöördvandefunktsioon is related to questions about the existence and distribution of perfect numbers, abundant numbers, and deficient numbers, which are classified based on the value of $\sigma(m)$ relative to m. Finding solutions to $\sigma(m) = n$ can be computationally challenging, especially for large values of n, as it often involves testing potential values of m or using properties of the sum of divisors function, such as its multiplicative nature.