Knödel Numbers

For every $k\geq 1$, let $C_k$ be the set of Composite numbers $n>k$ such that if $1<a<n$, GCD$(a,n)=1$ (where GCD is the Greatest Common Divisor), then $a^{n-k}\equiv 1\ \left({{\rm mod\ } {n}}\right)$. $C_1$ is the set of Carmichael Numbers. Makowski (1962/1963) proved that there are infinitely many members of $C_k$ for $k\geq 2$.

See also Carmichael Number, D-Number, Greatest Common Divisor

References

Makowski, A. ``Generalization of Morrow's $D$-Numbers.'' Simon Stevin 36, 71, 1962/1963.

Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 101, 1989.