Zeisel number

A Zeisel number, named after Helmut Zeisel, is a square-free integer k with at least three prime factors which fall into the pattern


 * $$p_x = ap_{x - 1} + b$$

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, $$p_0 = 1$$. 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



\begin{align} p_1 = 7, & {}\quad p_1 = 1p_0 + 6 \\ p_2 = 13, & {}\quad p_2 = 1p_1 + 6 \\ p_3 = 19, & {}\quad p_3 = 1p_2 + 6 \end{align}$$

1729 is an example for Carmichael numbers of the kind $$(6n + 1)(12n + 1)(18n + 1)$$, which satisfies the pattern $$p_x = ap_{x - 1} + b$$ 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, ….

The name Zeisel numbers was probably introduced by Kevin Brown, who was looking for numbers that when plugged into the equation


 * $$2^{k - 1} + k$$

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.

Hardy–Ramanujan's number 1729 is also a Zeisel number.