Revision 135746482 of "Zeisel number" on enwiki

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 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>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 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.

==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:Integer sequences]]

[[fr:Nombre de Zeisel]]
[[fi:Zeiselin luku]]