A Zeisel number is a square-free integer k with at least three prime factors which fall into the pattern
- px = apx - 1 + b
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, p0 = 1. The first few Zeisel numbers are
105, 1419, 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
- p1 = 7,p1 = 1p0 + 6
- p2 = 13,p2 = 1p1 + 6
- p3 = 19,p3 = 1p2 + 6
1729 is an example for Carmichael numbers of the kind (6n+1)(12n+1)(18n+1), which satisfied the pattern px = apx - 1 + b 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 Carmichel 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
- 2k - 1 + k
yield prime numbers. 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.
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