Zeisel number

A Zeisel number is a square-free integer N = p1 p2 ... pn such as the pis are distinct prime numbers and n ≥ 3 such that pi = A pi-1 + B for i = 1, 2, ..., n, p0 taken as 1, and with A and B some fixed integers. They are named after Helmut Zeisel, an Austrian mathematician.

The first few Zeisel numbers are:

105, 1419, 1729, 1885, 4505, 5719, 15387, 24211, 25085, 27559, 31929, 54205, 59081, 114985, 207177, 208681, 233569, ... (sequence A051015 in the OEIS)

These correspond to the mathematical constants: (1, 2), (4, −1), (1, 6) (2, 3), (3, 2), ...

History

Zeisel numbers were studied by Kevin Brown, who searched for prime solutions to 2n - 1 + n. Helmut Zeisel discovered that 1885 = n is prime for 2n - 1 + n. Brown discovered that its prime factors fit the recurrrence relation with A = 2 and B = 3. He named numbers fitting the recurrence relation "Zeisel numbers" which is now commonly used. Zeisel himself suggested the term "Brown-Zeisel number" because of their study of the number, but this has not caught on.

References

  1. 1 2 3 Gyllenbok, Jan (2018-04-11). "Zeisel number". Encyclopaedia of Historical Metrology, Weights, and Measures: Volume 1. Birkhäuser. pp. 236–237. ISBN 978-3-319-57598-8.
  2. 1 2 Weisstein, Eric W. "Zeisel Number". MathWorld. Wolfram Research, Inc. Retrieved 2026-10-28.
  3. 1 2 3 CompositeFan. "Zeisel number". PlanetMath. Retrieved 2026-10-28.
  4. 1 2 Weisstein, Eric W. (2002-12-12). "Zeisel Number". CRC Concise Encyclopedia of Mathematics. CRC Press. p. 3232. ISBN 978-1-4200-3522-3.