A Zeisel number, named after Helmut Zeisel, is a square-free integer

Square-free integer

In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...

 k with at least three prime factor

Prime factor

In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...

s which fall into the pattern


where a and b are some integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

 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, . The first few Zeisel numbers are

105

105 (number)

105 is the natural number following 104 and preceding 106.-In mathematics:105 is a triangular number, a 12-gonal number and a Zeisel number. It is a sphenic number, and is the product of three consecutive prime numbers. 105 is the double factorial of 7...

, 1419, 1729

1729 (number)

1729 is the natural number following 1728 and preceding 1730.1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan...

, 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


1729 is an example for Carmichael numbers of the kind , which satisfies the pattern 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


yield prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s. In a posting to the newsgroup

Newsgroup

A usenet newsgroup is a repository usually within the Usenet system, for messages posted from many users in different locations. The term may be confusing to some, because it is usually a discussion group. Newsgroups are technically distinct from, but functionally similar to, discussion forums on...

 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.

External links

• Wikisource:Zeisel numbers
• MathPages article