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# Zeisel numbers

The Zeisel numbers are squarefree numbers with at least three prime factors constructed in the following way: to construct a Zeisel number, you start with ${\displaystyle \scriptstyle p_{0}\,=\,1\,}$ and two integers ${\displaystyle \scriptstyle a\,}$ and ${\displaystyle \scriptstyle b\,}$. All numbers of the form ${\displaystyle \scriptstyle p_{n}\,=\,ap_{n-1}+b\,}$ with ${\displaystyle \scriptstyle n\,\geq \,1\,}$ have to be prime numbers.

The Zeisel numbers are named after the austrian mathematician Helmut Zeisel.

Examples:

${\displaystyle a=1,\,b=6:\ p_{1}=7,\,p_{2}=13,\,p_{3}=19\,}$ => ${\displaystyle 1729=7\cdot 13\cdot 19\,}$

${\displaystyle a=4,\,b=3:\ p_{1}=7,\,p_{2}=31,\,p_{3}=127\,}$ => ${\displaystyle 27559=7\cdot 31\cdot 127\,}$
${\displaystyle a=8,\,b=-3:\ p_{1}=5,\,p_{2}=37,\,p_{3}=293\,}$ => ${\displaystyle 54205=5\cdot 37\cdot 293\,}$

## Chernick Carmichael numbers and Zeisel numbers

Every Chernick Carmichael number is a Zeisel number with ${\displaystyle \scriptstyle a\,=\,1\,}$ and ${\displaystyle \scriptstyle b\,=\,6n\,}$.

## Generalization of the Zeisel numbers

It is possible to use a ${\displaystyle \scriptstyle p_{0}\,}$ dfferent from ${\displaystyle \scriptstyle 1\,}$

Examples:

• ${\displaystyle p_{0}=4,\ a=2,\ b=5}$
p0 = 4
p1 = a·p0 + b = 2·4  + 5 = 13
p2 = a·p1 + b = 2·13 + 5 = 31
p3 = a·p2 + b = 2·31 + 5 = 67

z = p1 · p2 · p3 = 13 · 31 · 67 = 27001

• ${\displaystyle p_{0}=-1,\ a=8,\ b=27}$
p0 = -1
p1 = a·p0 + b = 8·-1   + 27 =   19
p2 = a·p1 + b = 8·19   + 27 =  179
p3 = a·p2 + b = 28·179 + 27 = 1459

z = p1 · p2 · p3 = 19 · 179 · 1459 = 4962059


## Zeisel numbers and Fermat pseudoprimes

Every Zeisel number ${\displaystyle \scriptstyle n\,}$ is a Fermat pseudoprime to some base ${\displaystyle \scriptstyle b\,}$.

## Sequences

### Zeisel numbers related sequences

The Zeisel numbers (Cf. A051015) are

{105, 1419, 1729, 1885, 4505, 5719, 15387, 24211, 25085, 27559, 31929, 54205, 59081, 114985, 207177, 208681, 233569, 287979, 294409, 336611, 353977, 448585, 507579, 721907, 982513, ...}

The extended Chernick Carmichael numbers (Cf. AXXXXXX) are

{1729, 63973, 294409, ...}