The first one is from greedy , second is from takenouchi. Output of Graham Jewett looks like this
>>> egypt_fraction(Rational(7,12),"Graham Jewett")
[12, 13, 14, 15, 16, 17, 18, 156, 157, 158, 159, 160, 161, 182, 183, 184, 185, 186, 210, 211, 212, 213, 240, 241, 242, 272, 273, 306, 24492, 24493, 24494, 24495, 24496, 24806, 24807, 24808, 24809, 25122, 25123, 25124, 25440, 25441, 25760, 33306, 33307, 33308, 33309, 33672, 33673, 33674, 34040, 34041, 34410, 44310, 44311, 44312, 44732, 44733, 45156, 57840, 57841, 58322, 74256, 599882556, 599882557, 599882558, 599882559, 599931542, 599931543, 599931544, 599980530, 599980531, 600029520, 615362442, 615362443, 615362444, 615412056, 615412057, 615461672, 631140006, 631140007, 631190252, 647219040, 1109322942, 1109322943, 1109322944, 1109389556, 1109389557, 1109456172, 1133837256, 1133837257, 1133904602, 1158755640, 1963420410, 1963420411, 1963509032, 2000996556, 3345523440, 359859081592975692, 359859081592975693, 359859081592975694, 359859082792740806, 359859082792740807, 359859083992505922, 359917855686429306, 359917855686429307, 359917856886292392, 359976636979061430, 378670935639565806, 378670935639565807, 378670936870290692, 378731999285559192, 398337707804820042, 1230597390756858306, 1230597390756858307, 1230597392975504192, 1230745188071266692, 1285586924227446792, 3855019708367988510, 129498558604939936868397356895854556, 129498558604939936868397356895854557, 129498558604939937588115520081805942, 129498559468432682410497252290270442, 129540862841917352680420307822070942, 143391677498144189291812575843995442, 1514369938137587813730274566118047942, 16769876680757063368089314196389622249367851612542961252860614401811692]