A Zeisel number is a square-free integer k with at least three prime factors which fall into the pattern In mathematics, a square-free integer is one divisible by no perfect square, except 1. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ...

p_x = ap_{x − 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

(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 105 (one hundred [and] five) is the natural number following 104 and preceding 106. ... This article is about the number 1729. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...

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 p_x = ap_{x − 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. In number theory, a Carmichael number is a composite positive integer n which satisfies the congruence bn − 1 ≡ 1 (mod n) for all integers b which are relatively prime to n (see modular arithmetic). ...

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

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. In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... A newsgroup is a repository, usually within the Usenet system, for messages posted from many users at different locations. ...

External links

• Sloane Sequence A051015
• Wikisource:Zeisel_numbers
• MathWorld article
• MathPages article