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A '''Zeisel number''', named after [[Helmut Zeisel]], is a [[square-free integer]] ''k'' with at least three [[prime factor]]s which fall into the pattern
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:<math>p_x = ap_{x - 1} + b</math>
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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, <math>p_0 = 1</math>. The first few Zeisel numbers are
 
:[[105 (number)|105]], 1419, [[1729 (number)|1729]], 1885, 4505, 5719, 15387, 24211, 25085, 27559, 31929, 54205, 59081, 114985, 207177, 208681, 233569, 287979, 294409, 336611, 353977, 448585, 507579, 982513, 1012121, 1073305, 1242709, 1485609, 2089257, 2263811, 2953711, … {{OEIS|id=A051015}}.
 
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
 
:<math>
\begin{align}
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{align}</math>
 
1729 is an example for [[Carmichael number]]s of the kind <math>(6n + 1)(12n + 1)(18n + 1)</math>, which satisfies the pattern <math>p_x = ap_{x - 1} + b</math> with ''a''= 1 and ''b'' = 6n, so that every Carmichael number of the form (6n+1)(12n+1)(18n+1) is a Zeisel number.
 
Other Carmichael numbers of that kind are: 294409, 56052361, 118901521, 172947529, 216821881, 228842209, 1299963601, 2301745249, 9624742921, …
 
The name Zeisel numbers was probably introduced by Kevin Brown, who was looking for numbers that when plugged into the equation
 
:<math>2^{k - 1} + k</math>
 
yield [[prime number]]s. 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.
 
Hardy Ramanujan's number 1729 is also a Zeisel number.
 
==Notes==
{{reflist}}
 
==External links==
*[[Wikisource:Zeisel numbers]]
*{{MathWorld|urlname=ZeiselNumber|title=Zeisel Number}}
*[http://www.mathpages.com/home/kmath015.htm MathPages article]
 
{{Classes of natural numbers}}
 
[[Category:Integer sequences]]

Latest revision as of 18:47, 14 December 2014

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