Zeisel number

Given a squarefreeMathworldPlanetmath integer


with ω(n)>2 (in which ω(n) is number of distinct prime factors function, and all the pi are prime divisorsPlanetmathPlanetmath of n, except p0=1 purely as a notational convenience, and are sorted in ascending order) if each prime pi fits into the recurrence relation pi=mpi-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=1×3×11×43. 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 numbersMathworldPlanetmath 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 2n-1+n. Helmut Zeisel replied that 1885 is such an n. Brown discovered that its prime factors fit the recurrrence relationMathworldPlanetmath 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.

Title Zeisel number
Canonical name ZeiselNumber
Date of creation 2013-03-22 17:44:03
Last modified on 2013-03-22 17:44:03
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 6
Author CompositeFan (12809)
Entry type Definition
Classification msc 11A25
Defines Brown-Zeisel number