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Zeisel numbers
From OeisWiki
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 and two integers and . All numbers of the form with have to be prime numbers.
The Zeisel numbers are named after the austrian mathematician Helmut Zeisel.
Examples:
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Contents
Chernick Carmichael numbers and Zeisel numbers
Every Chernick Carmichael number is a Zeisel number with and .
Generalization of the Zeisel numbers
It is possible to use a dfferent from
Examples:
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
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 is a Fermat pseudoprime to some base .
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, ...}
See also
- Wikibooks (Deutsch), Pseudoprimzahlen: Zeisel-Zahlen.