Zeisel number

Given a squarefree integer

$$n=\prod _{i=0}^{\omega (n)}{p}_i,$$

with $\omega (n)>2$ (in which $\omega (n)$ is number of distinct prime factors function, and all the $p_i$ are prime divisors of $n$, except $p_0=1$ purely as a notational convenience, and are sorted in ascending order) if each prime $p_i$ fits into the recurrence relation $p_i=m{p}_{i-1}+a$, with $m$ being some fixed integer multiplicand, and $a$ being some fixed integer addend, then $n$ is called a Zeisel number.

For example, $1419=1\times 3\times 11\times 43$. Say that $m=4$ and $a=-1$. This checks out: $3=m+a$, $11=3m+a$ and $43=11m+a$. 1419 is a Zeisel number. The first few Zeisel numbers are 105, 1419, 1729, 1885, 4505, 5719, … listed in A051015 of Sloane’s OEIS. The Carmichael numbers of the form $(6n+1)(12n+1)(18n+1)$ are a subset of the Zeisel numbers; the constants are then $m=1$ and $a=6n$.

These numbers were first studied by Kevin Brown, who was searching for prime solutions to $2^{n-1}+n$. Helmut Zeisel replied that 1885 is such an $n$. Brown discovered that its prime factors fit the recurrrence relation with $m=2$ and $a=3$. He called numbers fitting such a recurrence relation “Zeisel numbers” and the term has stuck, being taken up by the OEIS, MathWorld and Wikipedia. Zeisel himself has suggested the term “Brown-Zeisel number” but this has not caught on. There is a different concept of the Zeisel number used in chemistry.