Knödel Numbers

For every , let  be the set of composite numbers  such that if ,  (where GCD is the greatest common divisor), then .

Special cases include , which is the set of Carmichael numbers, and , which gives the D-numbers.

Makowski (1962/1963) proved that there are infinitely many members of  for . The following table summarized Knödel numbers  for small .

	OEIS
1	A002997	561, 1105, 1729, 2465, 2821, 6601, 8911, ...
2	A050990	4, 6, 8, 10, 12, 14, 22, 24, 26, 30, ...
3	A033553	9, 15, 21, 33, 39, 51, 57, 63, 69, 87, ...
4	A050992	6, 8, 12, 16, 20, 24, 28, 40, 44, 48, ...
5	A050993	25, 65, 85, 145, 165, 185, 205, ...

REFERENCES:

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

Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 125-126, 1989.

Sloane, N. J. A. Sequences A002997/M5462, A033553, A050990, A050992, and A050993 in "The On-Line Encyclopedia of Integer Sequences."