Number theoretic aspects of a combinatorial function

L. Halbeisen and N. Hungerbühler
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 5, 1999, Number 4, Pages 138–150
Full paper (PDF, 434 Kb)

Details

Authors and affiliations

L. Halbeisen
Dep. of Mathematics
U.C. Berkeley Evans Hall 938
Berkeley, CA 94720 USA

N. Hungerbühler
Max-Planck Institute for Mathematics in the Sciences
Inselstrasse 22-26 04103 Leipzig Germany

Abstract

We investigate number theoretic aspects of the integer sequence A000522 of Sloane’s On-Line Encyclopedia of Integer Sequences. This integer sequence counts the number of sequences without repetition one can build with n distinct objects. By introducing the notion of the “shadow” of an integer function – which is related to its divisors – we treat some number theoretic properties
of this combinatorial function and investigate the related “irregular prime
numbers”.

References

1. J. M. Gandhi: On logarithmic numbers. Math. Student 31 (1963), 73-83.
2. L. HALBEISEN AND S. Shelah: Consequences of arithmetic for set theory. J. Symb. Logic 59 (1994), 30-40.
3. W. OBERSCHELP: Solving linear recurrences from differential equations in the exponential manner and vice versa, in “Applications of Fibonacci numbers, Vol. 6,” (G.E. Bergum, A.N. Philippou and A. F. Horadam, Ed.), 365-380, Kluwer Acad. Publ., (Dordrecht), 1996.
4. J. RlORDAN: “An Introduction to Combinatorial Analysis.” Princeton University Press, Princeton, New Jersey (1980).
5. D. Singh: The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers. Math. Student 20 (1952), 66-70.
6. N. J. A. Sloane: “A Handbook of Integer Sequences.” Academic Press, New York (1973).
7. N.J. A. Sloane and S. Plouffe: “The Encyclopedia of Integer Sequences.” Academic Press, San Diego (1995).