Using standard statistical definitions, the variance of $ x_1, x_2, ldots, x_n $ and squared errors on his mean $ mu $ are given by $ sigma ^ 2 = sum_i (x_i – mu) ^ 2 / n $ and $ delta_2 = sum_i (x_i – mu) ^ 2 = n sigma ^ 2 $ respectively.
Definition 1: The variance and squared error of an integer are defined as the variance and squared error of its positive divisors.
I found it:
- There are distinct whole pairs whose variances are equal. The smallest pair is $ (691, 817) $. Let's call them whole equivares.
- There are whole pairs whose squared errors are equal. The smallest pair is $ (45, 53) $.
The most interesting fact is that there are equivalent pairs that have the same number of divisors and, therefore, their squared errors are also equal. We define:
Definition 2: Two distinct integers are considered as an intimate pair if they have the same number of divisors and the same
The first intimate pairs are $ (1403,1461) $, $ (1564,1572) $,$ (2068,2076) $,$ (2249,2305) $,$ (3397,3493) $,$ (7871.8193) $,$ (23903,24101) $,$ (61769, 64443) $.
- Are there an infinity of intimate pairs?
- Are there three or more whole numbers that are intimate (and how should we call them hahaha)?
Note: This question was posted in MSE but not answered. Let me know if there is a reference in the literature. I did not find any.