Using the standard statistical definitions, the variance of $x_1, x_2, \ldots, x_n$ and the squared errors about its 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 the squared error of an integer is defined as the variance and the squared error of its positive divisors respectively.
I found that:
- There are distinct integer pairs whose variances are equal. The smallest such pair is $(691, 817)$. Let us call them equivarient integers.
- There are integer pairs whose squared errors are equal. The smallest such pair is $(45, 53)$.
The more interesting fact is that there are equivarient pairs which have the same number of divisors and hence their squared errors are also equal. We define:
Definition 2: Two distinct integers are said to be an intimate pair if they are have the same number of divisors and the same variance.
The first few intimate pairs are $(1403,1461)$, $(1564,1572)$,$(2068,2076)$,$(2249,2305)$,$(3397,3493)$,$(7871,8193)$,$ (23903,24101)$,$(61769, 64443)$.
- Are there infinitely many intimate pairs?
- Are there three or more integers that are intimate ( and what should we call them hahaha)
Note: Please let me know in case there is any reference in literature. I could not find any. Posted to MO since it is unanswered in MSE