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.

62 MAD numbers below 10^9 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 #31: 1657009 (2^27+1)/81 #32: 1838599 (2^27-1)/73 #33: 1965379 (2^36-1)/34965 #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=10^7 #43: 13788017 (2^33-1)/623 #44: 15790321 (2^28+1)/17 #45: 19173961 (2^27-1)/7 #46: 21225581 (2^42+1)/207205 #47: 24214051 (2^35+1)/1419 #48: 25080101 (2^34+1)/685 #49: 25781083 (2^37+1)/5331 #50: 53353631 (2^33-1)/161 #51: 102964687 (2^45-1)/341713 #52: 120296677 (2^38+1)/2285 #53: 164511353 (2^41-1)/13367 #54: 207207011 (2^45+1)/169803 #55: 240068041 (2^38+1)/1145 #56: 256957153 (2^45-1)/136927 #57: 464955857 (2^46+1)/151345 #58: 598781009 (2^42+1)/7345 #59: 616318177 (2^37-1)/223 #60: 715827883 (2^31+1)/3 #61: 905040953 (2^43-1)/9719 #62: 993089953 (2^54+1)/18139745 complete up to n=10^9 for pmorder < 200

Aart has some unpublished theory. It implies that (2^m-1)/d and (2^m+1)/d are mad when these are integral and d is sufficiently small. If such numbers are also prime, then they are MAD. For example, (2^43+1)/3 = 2932031007403 is MAD, and so is every Mersenne prime larger than 7, and every Fermat prime. There are infinitely many MAD numbers.

See also the very similar list for the Lights Out! version of this game. There is also information on a rectangular board.