Knödel number

In number theory, an n-Knödel number for a given positive integer n is a composite number m with the property that each i < m coprime to m satisfies ${\displaystyle i^{m-n}\equiv 1{\pmod {m}}}$.^[1] The concept is named after Walter Knödel.^{[citation needed]}

The set of all n-Knödel numbers is denoted K_n.^[1] The special case K₁ is the Carmichael numbers.^[1] There are infinitely many n-Knödel numbers for a given n.

Due to Euler's theorem every composite number m is an n-Knödel number for ${\displaystyle n=m-\varphi (m)}$ where ${\displaystyle \varphi }$ is Euler's totient function.

Examples

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