Zeisel number

A Zeisel number, named after Helmut Zeisel, is a square-free integer k with at least three prime factors which fall into the pattern

where a and b are some integer constants and x is the index number of each prime factor in the factorization, sorted from lowest to highest. For the purpose of determining Zeisel numbers, ${\displaystyle p_{0}=1}$. The first few Zeisel numbers are

To give an example, 1729 is a Zeisel number with the constants a = 1 and b = 6, its factors being 7, 13 and 19, falling into the pattern

1729 is an example for Carmichael numbers of the kind ${\displaystyle (6n+1)(12n+1)(18n+1)}$, which satisfies the pattern ${\displaystyle p_{x}=ap_{x-1}+b}$ with a= 1 and b = 6n, so that every Carmichael number of the form (6n+1)(12n+1)(18n+1) is a Zeisel number.

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