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# Knödel number

A Knödel number[1] for a given positive integer n is a composite number m with the property that each i < m coprime to m satisfies $i^{m - n} \equiv 1 \pmod{m}$. The set of all such integers for n is then called the set of Knödel numbers Kn.

The special case K1 are the Carmichael numbers.

Examples

n Kn
1 {561, 1105, 1729, 2465, 2821, 6601, ... } (sequence A002997 in OEIS)
2 {4, 6, 8, 10, 12, 14, 22, 24, 26, ... } (sequence A050990 in OEIS)
3 {9, 15, 21, 33, 39, 51, 57, 63, 69, ... } (sequence A033553 in OEIS)
4 {6, 8, 12, 16, 20, 24, 28, 40, 44, ... } (sequence A050992 in OEIS)

Literature

Makowski, A (1963). Generalization of Morrow's D-Numbers. p. 71.
Ribenboim, Paulo (1989). The New Book of Prime Number Records. New York: Springer-Verlag. p. 101. ISBN 9780387944579.
Weisstein, Eric W., "Knödel Numbers" from MathWorld.

References

^ Named after Walter Knödel, born May 20th, 1926 in Vienna. Knödel earned a Ph.D. in number theory in 1948 (advisors: Hlawka and Radon) and obtained the habilitation in 1953. Since 1961 he is professor at University of Stuttgart, establishing the new department of computer science, see also [1].