# nt.number theory – Is there an infinite number of intimate pairs of integers?

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.
respectively.

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
variance.

The first intimate pairs are $$(1403,1461)$$, $$(1564,1572)$$,$$(2068,2076)$$,$$(2249,2305)$$,$$(3397,3493)$$,$$(7871.8193)$$,$$(23903,24101)$$,$$(61769, 64443)$$.

Questions:

1. Are there an infinity of intimate pairs?
2. 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.