A Zeisel number, named after

Helmut Zeisel Helmut is a German name. Variants include Hellmut, Helmuth, and Hellmuth. From old German, the first element deriving from either ''heil'' ("healthy") or ''hiltja'' ("battle"), and the second from ''muot'' ("spirit, mind, mood"). Helmut may refer ...

, is a

square-free integer In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-f ...

''k'' with at least three

prime factor A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

s which fall into the pattern :

p_x = ap_ + b

where ''a'' and ''b'' are some

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

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 :

105 105 may refer to: *105 (number), the number *AD 105, a year in the 2nd century AD *105 BC, a year in the 2nd century BC * 105 (telephone number) * 105 (MBTA bus) * 105 (Northumberland) Construction Regiment, Royal Engineers, an English military unit ...

, 1419,

1729 Events January–March * January 8 – Frederick, the eldest son of King George II of Great Britain is made Prince of Wales at the age of 21, a few months after he comes to Britain for the first time after growing up in Hanover ...

, 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
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

1729 is an example for

Carmichael number In number theory, a Carmichael number is a composite number n, which in modular arithmetic satisfies the congruence relation: :b^n\equiv b\pmod for all integers b. The relation may also be expressed in the form: :b^\equiv 1\pmod. for all integers ...

s of the kind

(6n + 1)(12n + 1)(18n + 1)

, which satisfies the pattern

p_x = ap_ + 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

yield

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

s. In a posting to the

newsgroup A Usenet newsgroup is a repository usually within the Usenet system, for messages posted from users in different locations using the Internet. They are discussion groups and are not devoted to publishing news. Newsgroups are technically distinct ...

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

is also a Zeisel number.

Notes

External links

*
MathPages article
{{Classes of natural numbers Integer sequences