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Zeisel number (Definition)

Given a squarefree integer

$\displaystyle n = \prod_{i = 0}^{\omega(n)} p_i,$
with $ \omega(n) > 2$ (in which $ \omega(n)$ is number of distinct prime factors function, and all the $ p_i$ are prime divisors of $ n$, except $ p_0 = 1$ purely as a notational convenience, and are sorted in ascending order) if each prime $ p_i$ fits into the recurrence relation $ p_i = mp_{i - 1} + a$, with $ m$ being some fixed integer multiplicand, and $ a$ being some fixed integer addend, then $ n$ is called a Zeisel number.

For example, $ 1419 = 1 \times 3 \times 11 \times 43$. Say that $ m = 4$ and $ a = -1$. This checks out: $ 3 = m + a$, $ 11 = 3m + a$ and $ 43 = 11m + a$. 1419 is a Zeisel number. The first few Zeisel numbers are 105, 1419, 1729, 1885, 4505, 5719, ... listed in A051015 of Sloane's OEIS. The Carmichael numbers of the form $ (6n + 1)(12n + 1)(18n + 1)$ are a subset of the Zeisel numbers; the constants are then $ m = 1$ and $ a = 6n$.

These numbers were first studied by Kevin Brown, who was searching for prime solutions to $ 2^{n - 1} + n$. Helmut Zeisel replied that 1885 is such an $ n$. Brown discovered that its prime factors fit the recurrrence relation with $ m = 2$ and $ a = 3$. He called numbers fitting such a recurrence relation “Zeisel numbers” and the term has stuck, being taken up by the OEIS, MathWorld and Wikipedia. Zeisel himself has suggested the term “Brown-Zeisel number” but this has not caught on. There is a different concept of the Zeisel number used in chemistry.



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Also defines:  Brown-Zeisel number
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Cross-references: Wikipedia, MathWorld, term, relation, prime factors, solutions, numbers, subset, Carmichael numbers, OEIS, fixed, recurrence relation, prime, ascending order, prime divisors, number of distinct prime factors function, integer, squarefree

This is version 3 of Zeisel number, born on 2008-01-11, modified 2008-01-15.
Object id is 10183, canonical name is ZeiselNumber.
Accessed 624 times total.

Classification:
AMS MSC: 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
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