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Zeisel number
A '''Zeisel number''' is a [[square-free]] [[integer]] ''k'' with at least three [[prime factor]]s which fall into the pattern 

:<math>p_x = ap_{x - 1} + b</math>

where ''a'' and ''b'' are fixed 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>p_1 = 7, p_1 = 1p_0 + 6</math>
:<math>p_2 = 13, p_2 = 1p_1 + 6</math>
:<math>p_3 = 19, p_3 = 1p_2 + 6</math>

1729 is an example for [[Carmichael number]]s of the kind (6n+1)(12n+1)(18n+1), which satisfied the pattern <math>p_x = ap_{x - 1} + b</math> with ''a''= 1 and ''b'' = 6n, so that every Carmichael number, you can construct with the formula (6n+1)(12n+1)(18n+1), is a Zeisel number.

Other Carmichael Nnumbers of that kind are: 294409, 56052361, 118901521, 172947529, 216821881, 228842209, 1299963601, 2301745249, 9624742921, ...

The name Zeisel Nnumbers 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.

==External links==

*[http://www.research.att.com/projects/OEIS?Anum=A051015 Sloane Sequence A051015]
*[[Wikisource:Zeisel_numbers]]
*[http://mathworld.wolfram.com/ZeiselNumber.html MathWorld article]
*[http://www.mathpages.com/home/kmath015.htm MathPages article]

[[Category:Number sequences]]

[[de:Zeisel-Zahl]]
[[fr:Nombre de Zeisel]]