Ramanujan's taxicab number and its ilk (Q6579275)

The taxicab number 1729 has the property (as Ramanujan famously observed) that it is the (smallest) sum of two cubes of positive integers in two different ways. It is also a Carmichael number. The author in this short paper gives a list of integers \(<10^{21}\) that share these properties. It is not known whether the list is finite or infinite. Also, he gives a list of Carmichael numbers that are of the form \(b^n\pm 1\) and gives a possible explanation pointing to the fact that Carmichael numbers represented by \(b^n - 1\) are rare. He finally studies the same problem for Lucas-Carmichael numbers.