Any multiple of a mad number is again mad, so the interesting ones are the mad numbers without proper mad divisors. Let us call them MAD.
45 MAD numbers are known, namely
#1: 3 (2^1+1) #2: 5 (2^2+1) #3: 17 (2^4+1) #4: 31 (2^5-1) #5: 127 (2^7-1) #6: 257 (2^8+1) #7: 511 (2^9-1) #8: 683 (2^11+1)/3 #9: 2047 (2^11-1) #10: 2731 (2^13+1)/3 #11: 3277 (2^14+1)/5 #12: 3641 (2^15+1)/9 #13: 8191 (2^13-1) #14: 43691 (2^17+1)/3 #15: 52429 (2^18+1)/5 #16: 61681 (2^20+1)/17 #17: 65537 (2^16+1) #18: 85489 (2^26+1)/785 #19: 131071 (2^17-1) #20: 174763 (2^19+1)/3 #21: 178481 (2^23-1)/47 #22: 233017 (2^21+1)/9 #23: 253241 (2^26+1)/265 #24: 256999 (2^29-1)/2089 #25: 486737 (2^29-1)/1103 #26: 524287 (2^19-1) #27: 704093 (2^30+1)/1525 #28: 838861 (2^22+1)/5 #29: 1016801 (2^25+1)/33 #30: 1082401 (2^25-1)/31 complete up to n=1090000 #31: 1657009 (2^27+1)/81 #32: 1838599 (2^27-1)/73 #33: 1965379 (2^36-1)/34965 complete up to n=2000000 for pmorder < 2000. #34: 2304167 (2^29-1)/233 #35: 2796203 (2^23+1)/3 #36: 3033169 (2^29+1)/177 #37: 3303821 (2^30+1)/325 #38: 3605429 (2^34+1)/4765 #39: 3705353 (2^35+1)/9273 #40: 6700417 (2^32+1)/641 #41: 8727391 (2^35-1)/3937 #42: 9335617 (2^36+1)/7361 complete up to n=10000000 for pmorder < 100 #43: 13788017 (2^33-1)/623 #44: 15790321 (2^28+1)/17 #45: 19173961 (2^27-1)/7 complete up to n=20000000 for pmorder < 50
Aart has some unpublished theory.
There is also information on a rectangular board.