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."
