Difference between revisions 4738192 and 5364833 on enwiki

A '''Zeisel number''' is a [[square-free]] integer ''k'' with at least three prime factors 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

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>
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:<math>p_2 = 13, p_2 = 1p_1 + 6</math>
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:<math>p_3 = 19, p_3 = 1p_2 + 6</math>

These kind of numbers were first studied by the Austrian? mathematician [[Helmut Zeisel]], who was looking for numbers that when plugged into the equation <math>2^{k - 1} + k</math> yield [[prime number]]s. 1885 is one such number that does so, as well as having prime factors with the relationship described above.

==External Resourcelinks==

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