On primary Carmichael numbers

The primary Carmichael numbers were recently introduced as a special subset of the Carmichael numbers. A primary Carmichael number \(m\) has the unique property that \(s_p(m) = p\) holds for each prime factor \(p\), where \(s_p(m)\) is the sum of the base-\(p\) digits of \(m\). The first such number is Ramanujan's famous taxicab number \(1729\). Due to Chernick, all Carmichael numbers with three factors can be constructed by certain squarefree polynomials \(U_3(t) \in \mathbb{Z}[t]\), the simplest one being \(U_3(t) = (6t+1)(12t+1)(18t+1)\). We show that the values of any \(U_3(t)\) obey a special decomposition for all \(t \geq 2\) and besides certain exceptions also in the case \(t=1\). These cases further imply that if all three factors of \(U_3(t)\) are simultaneously odd primes, then \(U_3(t)\) is not only a Carmichael number, but also a primary Carmichael number. Together with the exceptional cases, all Carmichael numbers with three factors have at least the property that \(s_p(m) = p\) holds for the greatest prime factor \(p\) of \(m\). Subsequently, we show some connections to taxicab and polygonal numbers, involving the number \(1729\) as an example again.