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

$$p_{x}=ap_{x-1}+b,$$

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. To determine Zeisel numbers, $p_{0}=1$. The first few Zeisel numbers are

105, 1419, 1729, 1885, 4505, 5719, 15387, … (sequence A051015 in the OEIS).

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

${\begin{aligned}p_{1}=7,&{}\quad p_{1}=1p_{0}+6\\p_{2}=13,&{}\quad p_{2}=1p_{1}+6\\p_{3}=19,&{}\quad p_{3}=1p_{2}+6\end{aligned}}$

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

The Zeisel number was probably introduced by Kevin Brown, who was looking for numbers that when plugged into the equation

$$2^{k-1}+k,$$

yielding prime numbers. In a posting to the newsgroup sci.math on 1994-02-24, Helmut Zeisel pointed out that 1885 is one such number. Later it was discovered (by Kevin Brown?) that 1885 additionally has prime factors with the relationship described above, so a name like Brown-Zeisel Numbers might be more appropriate.

External links

• Weisstein, Eric W. "Zeisel Number". MathWorld.
• MathPages article

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