Zeisel number
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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 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, 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
(sequence A051015 in OEIS). 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 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
- 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.
