Difference between revisions 769317961 and 967128071 on enwiki

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Zeisel number
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

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

(contracted; show full)
\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, …
 {{OEIS|id=A033502}}.

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 (number)|1729]] is also a Zeisel number.

==Notes==
{{reflist}}

==External links==
*{{MathWorld|urlname=ZeiselNumber|title=Zeisel Number}}
*[http://www.mathpages.com/home/kmath015.htm MathPages article]

{{Classes of natural numbers}}

[[Category:Integer sequences]]