Zeisel numbers

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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 $p_0=1$ and two integers $a$ and $b$. All numbers of the form $p_n=a{p}_{n-1}+b$ with $n\ge 1$ have to be prime numbers.

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

Examples:

$a=1,b=6:p_1=7,p_2=13,p_3=19$ => $1729=7\cdot 13\cdot 19$

$a=4,b=3:p_1=7,p_2=31,p_3=127$ => $27559=7\cdot 31\cdot 127$

$a=8,b=-3:p_1=5,p_2=37,p_3=293$ => $54205=5\cdot 37\cdot 293$

Chernick Carmichael numbers and Zeisel numbers

Every Chernick Carmichael number is a Zeisel number with $a=1$ and $b=6n$.

Generalization of the Zeisel numbers

It is possible to use a $p_0$ dfferent from $1$

Examples:

• $p_0=4,a=2,b=5$

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 

• $p_0=-1,a=8,b=27$

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 $n$ is a Fermat pseudoprime to some base $b$.

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. OEIS:AXXXXXX) are

{1729, 63973, 294409, ...}

See also

• Wikibooks (Deutsch), Pseudoprimzahlen: Zeisel-Zahlen.