77 is the greatest integer that cannot be a finite sum of distinct integers greater than 1 whose sum of their reciprocals is 1
This program gave a definitive check for all numbers up to $$104 = 2+3+4+5+6+7+8+9+10+11+12+13+14.$$ As you can see, by that point there are multiple solutions. Also, no solutions for $...58,63,68,70, 72,77, \; $ but then success. Programming note: at first i added the reciprocals as double precision floats and accepted if within a very small error from $1.0$ Then I started to see sums that made me a little supicious, 0.9998 sort of thing, so i switched it to GMP and demanded that the sum of the product over each individual entry be the same as the product. No doubt that way. Note that, for those examples that work, the least common multiple od the numbers to be added is relatively small; I had the computer list that at the end of each line, less work for a human being to confirm one of these by hand (compared with the product).
11 2 3 6 LCM 6
24 2 4 6 12 LCM 12
30 2 3 10 15 LCM 30
31 2 4 5 20 LCM 20
32 2 3 9 18 LCM 18
37 2 3 8 24 LCM 24
38 3 4 5 6 20 LCM 60
43 2 4 10 12 15 LCM 60
45 2 4 9 12 18 LCM 36
45 2 5 6 12 20 LCM 60
50 2 4 8 12 24 LCM 24
50 3 4 6 10 12 15 LCM 60
52 3 4 6 9 12 18 LCM 36
53 2 5 6 10 30 LCM 30
54 2 3 7 42 LCM 42
55 2 4 7 14 28 LCM 28
57 3 4 5 10 15 20 LCM 60
57 3 4 6 8 12 24 LCM 24
59 3 4 5 9 18 20 LCM 180
60 2 6 9 10 15 18 LCM 90
61 2 4 6 21 28 LCM 84
62 2 4 6 20 30 LCM 60
62 3 4 6 7 14 28 LCM 84
64 2 4 8 10 40 LCM 40
64 2 5 10 12 15 20 LCM 60
64 3 4 5 10 12 30 LCM 60
64 3 4 5 8 20 24 LCM 120
65 2 6 8 10 15 24 LCM 120
66 2 3 12 21 28 LCM 84
66 2 4 6 18 36 LCM 36
66 2 5 9 12 18 20 LCM 180
67 2 3 12 20 30 LCM 60
67 2 4 7 12 42 LCM 84
67 2 5 6 9 45 LCM 90
67 2 6 8 9 18 24 LCM 72
69 2 3 14 15 35 LCM 210
69 2 6 7 12 14 28 LCM 84
71 2 3 11 22 33 LCM 66
71 2 3 12 18 36 LCM 36
71 2 5 8 12 20 24 LCM 120
71 3 4 6 8 10 40 LCM 120
71 3 4 9 10 12 15 18 LCM 180
71 3 5 6 10 12 15 20 LCM 60
73 3 5 6 9 12 18 20 LCM 180
74 2 5 9 10 18 30 LCM 90
74 3 4 6 7 12 42 LCM 84
75 3 4 5 8 15 40 LCM 120
76 2 4 6 16 48 LCM 48
76 2 5 7 14 20 28 LCM 140
76 3 4 8 10 12 15 24 LCM 120
78 2 6 8 10 12 40 LCM 120
78 3 4 5 9 12 45 LCM 180
78 3 4 8 9 12 18 24 LCM 72
78 3 5 6 8 12 20 24 LCM 120
79 2 3 10 24 40 LCM 120
79 2 5 8 10 24 30 LCM 120
80 2 4 10 15 21 28 LCM 420
81 2 3 12 16 48 LCM 48
81 2 4 10 15 20 30 LCM 60
81 3 4 5 7 20 42 LCM 420
81 3 4 7 10 14 15 28 LCM 420
81 3 5 6 9 10 18 30 LCM 90
82 2 4 12 14 15 35 LCM 420
82 2 4 9 18 21 28 LCM 252
82 2 5 6 20 21 28 LCM 420
82 2 5 8 12 15 40 LCM 120
82 2 6 7 10 15 42 LCM 210
83 2 4 9 18 20 30 LCM 180
83 3 4 7 9 14 18 28 LCM 252
83 3 5 6 7 14 20 28 LCM 420
84 2 4 11 12 22 33 LCM 132
84 2 6 7 9 18 42 LCM 126
85 2 4 10 14 20 35 LCM 140
85 2 4 10 15 18 36 LCM 180
85 2 5 8 10 20 40 LCM 40
86 2 5 9 10 15 45 LCM 90
86 2 8 9 10 15 18 24 LCM 360
86 3 5 6 8 10 24 30 LCM 120
87 2 4 6 15 60 LCM 60
87 2 4 8 21 24 28 LCM 168
87 2 5 6 18 20 36 LCM 180
87 3 4 6 10 15 21 28 LCM 420
87 4 5 6 9 10 15 18 20 LCM 180
88 2 4 8 20 24 30 LCM 120
88 2 5 7 12 20 42 LCM 420
88 2 7 10 12 14 15 28 LCM 420
88 3 4 6 10 15 20 30 LCM 60
88 3 4 7 8 14 24 28 LCM 168
89 2 3 9 30 45 LCM 90
89 2 6 7 8 24 42 LCM 168
89 3 4 6 12 14 15 35 LCM 420
89 3 4 6 9 18 21 28 LCM 252
89 3 5 6 8 12 15 40 LCM 120
90 2 7 9 12 14 18 28 LCM 252
90 3 4 6 9 18 20 30 LCM 180
91 3 4 6 11 12 22 33 LCM 132
92 2 3 12 15 60 LCM 60
92 2 4 10 12 24 40 LCM 120
92 2 4 5 36 45 LCM 180
92 2 4 8 18 24 36 LCM 72
92 2 5 6 14 30 35 LCM 210
92 2 5 6 15 24 40 LCM 120
92 3 4 6 10 14 20 35 LCM 420
92 3 4 6 10 15 18 36 LCM 180
92 3 4 8 9 10 18 40 LCM 360
92 3 5 6 8 10 20 40 LCM 120
92 3 5 9 10 12 15 18 20 LCM 180
92 4 5 6 8 10 15 20 24 LCM 120
93 2 5 8 9 24 45 LCM 360
93 3 4 5 12 20 21 28 LCM 420
93 3 4 7 10 12 15 42 LCM 420
93 3 5 6 9 10 15 45 LCM 90
93 3 6 8 9 10 15 18 24 LCM 360
94 2 6 10 12 15 21 28 LCM 420
94 3 4 6 8 21 24 28 LCM 168
94 4 5 6 8 9 18 20 24 LCM 360
94 4 5 6 9 10 12 18 30 LCM 180
95 2 3 10 20 60 LCM 60
95 2 3 9 27 54 LCM 54
95 2 4 10 15 16 48 LCM 240
95 2 4 8 9 72 LCM 72
95 2 4 9 15 20 45 LCM 180
95 2 6 10 12 15 20 30 LCM 60
95 2 7 8 12 14 24 28 LCM 168
95 3 4 6 8 20 24 30 LCM 120
95 3 4 7 9 12 18 42 LCM 252
95 3 5 6 7 12 20 42 LCM 420
95 3 6 7 10 12 14 15 28 LCM 420
96 2 5 7 10 30 42 LCM 210
96 2 6 9 12 18 21 28 LCM 252
96 3 4 5 14 15 20 35 LCM 420
96 4 5 6 7 12 14 20 28 LCM 420
97 2 4 9 16 18 48 LCM 144
97 2 5 6 16 20 48 LCM 240
97 2 6 9 12 18 20 30 LCM 180
97 3 5 8 10 12 15 20 24 LCM 120
97 3 6 7 9 12 14 18 28 LCM 252
98 3 4 5 11 20 22 33 LCM 660
98 3 4 5 12 18 20 36 LCM 180
99 2 4 8 15 30 40 LCM 120
99 2 6 10 11 15 22 33 LCM 330
99 2 6 10 12 14 20 35 LCM 420
99 2 6 10 12 15 18 36 LCM 180
99 2 6 9 14 15 18 35 LCM 630
99 2 8 9 10 12 18 40 LCM 360
99 3 4 5 6 36 45 LCM 180
99 3 4 6 10 12 24 40 LCM 120
99 3 4 6 8 18 24 36 LCM 72
99 3 5 8 9 12 18 20 24 LCM 360
99 4 5 6 8 10 12 24 30 LCM 120
100 2 6 7 8 21 56 LCM 168
100 3 4 7 8 12 24 42 LCM 168
100 3 5 6 8 9 24 45 LCM 360
101 2 4 5 30 60 LCM 60
101 2 5 10 15 20 21 28 LCM 420
101 2 6 8 12 21 24 28 LCM 168
101 2 6 9 11 18 22 33 LCM 198
101 3 4 5 10 21 28 30 LCM 420
102 2 4 8 16 24 48 LCM 48
102 2 4 9 12 30 45 LCM 180
102 2 6 8 12 20 24 30 LCM 120
102 3 4 6 10 15 16 48 LCM 240
102 3 4 6 8 9 72 LCM 72
102 3 4 6 9 15 20 45 LCM 180
102 3 5 7 10 14 15 20 28 LCM 420
102 3 6 7 8 12 14 24 28 LCM 168
103 2 4 8 14 35 40 LCM 280
103 2 5 12 14 15 20 35 LCM 420
103 2 5 9 18 20 21 28 LCM 1260
103 2 7 9 10 15 18 42 LCM 630
103 3 4 5 12 14 30 35 LCM 420
103 3 4 5 12 15 24 40 LCM 120
103 3 5 6 7 10 30 42 LCM 210
104 2 3 20 21 28 30 LCM 420
104 2 4 7 21 28 42 LCM 84
104 2 6 8 14 15 24 35 LCM 840
104 3 4 6 9 16 18 48 LCM 144
104 3 5 7 9 14 18 20 28 LCM 1260
104 4 5 6 7 10 14 28 30 LCM 420
105 2 4 7 20 30 42 LCM 420
105 2 5 11 12 20 22 33 LCM 660
105 4 5 6 8 10 12 20 40 LCM 120
105 4 5 6 8 9 15 18 40 LCM 360
106 2 5 10 15 18 20 36 LCM 180
106 2 5 6 12 36 45 LCM 180
106 2 6 8 11 22 24 33 LCM 264
106 2 6 8 12 18 24 36 LCM 72
106 3 4 5 10 18 30 36 LCM 180
106 3 4 5 10 20 24 40 LCM 120
106 3 4 6 8 15 30 40 LCM 120
106 3 6 8 9 10 12 18 40 LCM 360
106 4 5 6 9 10 12 15 45 LCM 180
106 4 6 8 9 10 12 15 18 24 LCM 360
107 2 6 7 14 20 28 30 LCM 420
107 3 5 8 9 10 18 24 30 LCM 360
108 2 3 18 21 28 36 LCM 252
108 2 4 10 12 20 60 LCM 60
108 2 4 9 12 27 54 LCM 108
108 2 4 9 15 18 60 LCM 180
108 2 5 10 12 21 28 30 LCM 420
108 2 5 6 15 20 60 LCM 60
108 2 5 8 20 21 24 28 LCM 840
108 2 7 8 10 15 24 42 LCM 840
108 3 4 5 12 16 20 48 LCM 240
108 3 4 5 6 30 60 LCM 60
108 3 4 9 10 15 18 21 28 LCM 1260
108 3 5 6 10 15 20 21 28 LCM 420
109 2 3 18 20 30 36 LCM 180
109 2 4 5 28 70 LCM 140
109 2 4 7 18 36 42 LCM 252
109 2 6 10 12 15 16 48 LCM 240
109 2 6 8 9 12 72 LCM 72
109 2 6 9 10 18 24 40 LCM 360
109 2 6 9 12 15 20 45 LCM 180
109 2 7 8 10 14 28 40 LCM 280
109 3 4 6 8 16 24 48 LCM 48
109 3 4 6 9 12 30 45 LCM 180
109 3 4 9 10 15 18 20 30 LCM 180
109 3 5 7 10 12 14 28 30 LCM 420
109 3 5 7 8 14 20 24 28 LCM 840
109 4 5 6 7 10 15 20 42 LCM 420
110 2 3 9 24 72 LCM 72
110 2 4 6 14 84 LCM 84
110 2 5 7 9 42 45 LCM 630
110 2 7 8 9 18 24 42 LCM 504
110 3 4 10 11 12 15 22 33 LCM 660
110 3 4 6 8 14 35 40 LCM 840
110 3 4 9 12 14 15 18 35 LCM 1260
110 3 5 6 12 14 15 20 35 LCM 420
110 3 5 6 9 18 20 21 28 LCM 1260
110 3 5 8 9 12 15 18 40 LCM 360
110 3 6 7 9 10 15 18 42 LCM 630
I had the numbers pushed close together with commas, then added up the LCM divided by each number. Seems to have come out perfectly, the sum should be the same as the LCM. I had to stop at 106, too many characters total otherwise.
11 2,3,6 LCM 6 3+2+1=6
24 2,4,6,12 LCM 12 6+3+2+1=12
30 2,3,10,15 LCM 30 15+10+3+2=30
31 2,4,5,20 LCM 20 10+5+4+1=20
32 2,3,9,18 LCM 18 9+6+2+1=18
37 2,3,8,24 LCM 24 12+8+3+1=24
38 3,4,5,6,20 LCM 60 20+15+12+10+3=60
43 2,4,10,12,15 LCM 60 30+15+6+5+4=60
45 2,4,9,12,18 LCM 36 18+9+4+3+2=36
45 2,5,6,12,20 LCM 60 30+12+10+5+3=60
50 2,4,8,12,24 LCM 24 12+6+3+2+1=24
50 3,4,6,10,12,15 LCM 60 20+15+10+6+5+4=60
52 3,4,6,9,12,18 LCM 36 12+9+6+4+3+2=36
53 2,5,6,10,30 LCM 30 15+6+5+3+1=30
54 2,3,7,42 LCM 42 21+14+6+1=42
55 2,4,7,14,28 LCM 28 14+7+4+2+1=28
57 3,4,5,10,15,20 LCM 60 20+15+12+6+4+3=60
57 3,4,6,8,12,24 LCM 24 8+6+4+3+2+1=24
59 3,4,5,9,18,20 LCM 180 60+45+36+20+10+9=180
60 2,6,9,10,15,18 LCM 90 45+15+10+9+6+5=90
61 2,4,6,21,28 LCM 84 42+21+14+4+3=84
62 2,4,6,20,30 LCM 60 30+15+10+3+2=60
62 3,4,6,7,14,28 LCM 84 28+21+14+12+6+3=84
64 2,4,8,10,40 LCM 40 20+10+5+4+1=40
64 2,5,10,12,15,20 LCM 60 30+12+6+5+4+3=60
64 3,4,5,10,12,30 LCM 60 20+15+12+6+5+2=60
64 3,4,5,8,20,24 LCM 120 40+30+24+15+6+5=120
65 2,6,8,10,15,24 LCM 120 60+20+15+12+8+5=120
66 2,3,12,21,28 LCM 84 42+28+7+4+3=84
66 2,4,6,18,36 LCM 36 18+9+6+2+1=36
66 2,5,9,12,18,20 LCM 180 90+36+20+15+10+9=180
67 2,3,12,20,30 LCM 60 30+20+5+3+2=60
67 2,4,7,12,42 LCM 84 42+21+12+7+2=84
67 2,5,6,9,45 LCM 90 45+18+15+10+2=90
67 2,6,8,9,18,24 LCM 72 36+12+9+8+4+3=72
69 2,3,14,15,35 LCM 210 105+70+15+14+6=210
69 2,6,7,12,14,28 LCM 84 42+14+12+7+6+3=84
71 2,3,11,22,33 LCM 66 33+22+6+3+2=66
71 2,3,12,18,36 LCM 36 18+12+3+2+1=36
71 2,5,8,12,20,24 LCM 120 60+24+15+10+6+5=120
71 3,4,6,8,10,40 LCM 120 40+30+20+15+12+3=120
71 3,4,9,10,12,15,18 LCM 180 60+45+20+18+15+12+10=180
71 3,5,6,10,12,15,20 LCM 60 20+12+10+6+5+4+3=60
73 3,5,6,9,12,18,20 LCM 180 60+36+30+20+15+10+9=180
74 2,5,9,10,18,30 LCM 90 45+18+10+9+5+3=90
74 3,4,6,7,12,42 LCM 84 28+21+14+12+7+2=84
75 3,4,5,8,15,40 LCM 120 40+30+24+15+8+3=120
76 2,4,6,16,48 LCM 48 24+12+8+3+1=48
76 2,5,7,14,20,28 LCM 140 70+28+20+10+7+5=140
76 3,4,8,10,12,15,24 LCM 120 40+30+15+12+10+8+5=120
78 2,6,8,10,12,40 LCM 120 60+20+15+12+10+3=120
78 3,4,5,9,12,45 LCM 180 60+45+36+20+15+4=180
78 3,4,8,9,12,18,24 LCM 72 24+18+9+8+6+4+3=72
78 3,5,6,8,12,20,24 LCM 120 40+24+20+15+10+6+5=120
79 2,3,10,24,40 LCM 120 60+40+12+5+3=120
79 2,5,8,10,24,30 LCM 120 60+24+15+12+5+4=120
80 2,4,10,15,21,28 LCM 420 210+105+42+28+20+15=420
81 2,3,12,16,48 LCM 48 24+16+4+3+1=48
81 2,4,10,15,20,30 LCM 60 30+15+6+4+3+2=60
81 3,4,5,7,20,42 LCM 420 140+105+84+60+21+10=420
81 3,4,7,10,14,15,28 LCM 420 140+105+60+42+30+28+15=420
81 3,5,6,9,10,18,30 LCM 90 30+18+15+10+9+5+3=90
82 2,4,12,14,15,35 LCM 420 210+105+35+30+28+12=420
82 2,4,9,18,21,28 LCM 252 126+63+28+14+12+9=252
82 2,5,6,20,21,28 LCM 420 210+84+70+21+20+15=420
82 2,5,8,12,15,40 LCM 120 60+24+15+10+8+3=120
82 2,6,7,10,15,42 LCM 210 105+35+30+21+14+5=210
83 2,4,9,18,20,30 LCM 180 90+45+20+10+9+6=180
83 3,4,7,9,14,18,28 LCM 252 84+63+36+28+18+14+9=252
83 3,5,6,7,14,20,28 LCM 420 140+84+70+60+30+21+15=420
84 2,4,11,12,22,33 LCM 132 66+33+12+11+6+4=132
84 2,6,7,9,18,42 LCM 126 63+21+18+14+7+3=126
85 2,4,10,14,20,35 LCM 140 70+35+14+10+7+4=140
85 2,4,10,15,18,36 LCM 180 90+45+18+12+10+5=180
85 2,5,8,10,20,40 LCM 40 20+8+5+4+2+1=40
86 2,5,9,10,15,45 LCM 90 45+18+10+9+6+2=90
86 2,8,9,10,15,18,24 LCM 360 180+45+40+36+24+20+15=360
86 3,5,6,8,10,24,30 LCM 120 40+24+20+15+12+5+4=120
87 2,4,6,15,60 LCM 60 30+15+10+4+1=60
87 2,4,8,21,24,28 LCM 168 84+42+21+8+7+6=168
87 2,5,6,18,20,36 LCM 180 90+36+30+10+9+5=180
87 3,4,6,10,15,21,28 LCM 420 140+105+70+42+28+20+15=420
87 4,5,6,9,10,15,18,20 LCM 180 45+36+30+20+18+12+10+9=180
88 2,4,8,20,24,30 LCM 120 60+30+15+6+5+4=120
88 2,5,7,12,20,42 LCM 420 210+84+60+35+21+10=420
88 2,7,10,12,14,15,28 LCM 420 210+60+42+35+30+28+15=420
88 3,4,6,10,15,20,30 LCM 60 20+15+10+6+4+3+2=60
88 3,4,7,8,14,24,28 LCM 168 56+42+24+21+12+7+6=168
89 2,3,9,30,45 LCM 90 45+30+10+3+2=90
89 2,6,7,8,24,42 LCM 168 84+28+24+21+7+4=168
89 3,4,6,12,14,15,35 LCM 420 140+105+70+35+30+28+12=420
89 3,4,6,9,18,21,28 LCM 252 84+63+42+28+14+12+9=252
89 3,5,6,8,12,15,40 LCM 120 40+24+20+15+10+8+3=120
90 2,7,9,12,14,18,28 LCM 252 126+36+28+21+18+14+9=252
90 3,4,6,9,18,20,30 LCM 180 60+45+30+20+10+9+6=180
91 3,4,6,11,12,22,33 LCM 132 44+33+22+12+11+6+4=132
92 2,3,12,15,60 LCM 60 30+20+5+4+1=60
92 2,4,10,12,24,40 LCM 120 60+30+12+10+5+3=120
92 2,4,5,36,45 LCM 180 90+45+36+5+4=180
92 2,4,8,18,24,36 LCM 72 36+18+9+4+3+2=72
92 2,5,6,14,30,35 LCM 210 105+42+35+15+7+6=210
92 2,5,6,15,24,40 LCM 120 60+24+20+8+5+3=120
92 3,4,6,10,14,20,35 LCM 420 140+105+70+42+30+21+12=420
92 3,4,6,10,15,18,36 LCM 180 60+45+30+18+12+10+5=180
92 3,4,8,9,10,18,40 LCM 360 120+90+45+40+36+20+9=360
92 3,5,6,8,10,20,40 LCM 120 40+24+20+15+12+6+3=120
92 3,5,9,10,12,15,18,20 LCM 180 60+36+20+18+15+12+10+9=180
92 4,5,6,8,10,15,20,24 LCM 120 30+24+20+15+12+8+6+5=120
93 2,5,8,9,24,45 LCM 360 180+72+45+40+15+8=360
93 3,4,5,12,20,21,28 LCM 420 140+105+84+35+21+20+15=420
93 3,4,7,10,12,15,42 LCM 420 140+105+60+42+35+28+10=420
93 3,5,6,9,10,15,45 LCM 90 30+18+15+10+9+6+2=90
93 3,6,8,9,10,15,18,24 LCM 360 120+60+45+40+36+24+20+15=360
94 2,6,10,12,15,21,28 LCM 420 210+70+42+35+28+20+15=420
94 3,4,6,8,21,24,28 LCM 168 56+42+28+21+8+7+6=168
94 4,5,6,8,9,18,20,24 LCM 360 90+72+60+45+40+20+18+15=360
94 4,5,6,9,10,12,18,30 LCM 180 45+36+30+20+18+15+10+6=180
95 2,3,10,20,60 LCM 60 30+20+6+3+1=60
95 2,3,9,27,54 LCM 54 27+18+6+2+1=54
95 2,4,10,15,16,48 LCM 240 120+60+24+16+15+5=240
95 2,4,8,9,72 LCM 72 36+18+9+8+1=72
95 2,4,9,15,20,45 LCM 180 90+45+20+12+9+4=180
95 2,6,10,12,15,20,30 LCM 60 30+10+6+5+4+3+2=60
95 2,7,8,12,14,24,28 LCM 168 84+24+21+14+12+7+6=168
95 3,4,6,8,20,24,30 LCM 120 40+30+20+15+6+5+4=120
95 3,4,7,9,12,18,42 LCM 252 84+63+36+28+21+14+6=252
95 3,5,6,7,12,20,42 LCM 420 140+84+70+60+35+21+10=420
95 3,6,7,10,12,14,15,28 LCM 420 140+70+60+42+35+30+28+15=420
96 2,5,7,10,30,42 LCM 210 105+42+30+21+7+5=210
96 2,6,9,12,18,21,28 LCM 252 126+42+28+21+14+12+9=252
96 3,4,5,14,15,20,35 LCM 420 140+105+84+30+28+21+12=420
96 4,5,6,7,12,14,20,28 LCM 420 105+84+70+60+35+30+21+15=420
97 2,4,9,16,18,48 LCM 144 72+36+16+9+8+3=144
97 2,5,6,16,20,48 LCM 240 120+48+40+15+12+5=240
97 2,6,9,12,18,20,30 LCM 180 90+30+20+15+10+9+6=180
97 3,5,8,10,12,15,20,24 LCM 120 40+24+15+12+10+8+6+5=120
97 3,6,7,9,12,14,18,28 LCM 252 84+42+36+28+21+18+14+9=252
98 3,4,5,11,20,22,33 LCM 660 220+165+132+60+33+30+20=660
98 3,4,5,12,18,20,36 LCM 180 60+45+36+15+10+9+5=180
99 2,4,8,15,30,40 LCM 120 60+30+15+8+4+3=120
99 2,6,10,11,15,22,33 LCM 330 165+55+33+30+22+15+10=330
99 2,6,10,12,14,20,35 LCM 420 210+70+42+35+30+21+12=420
99 2,6,10,12,15,18,36 LCM 180 90+30+18+15+12+10+5=180
99 2,6,9,14,15,18,35 LCM 630 315+105+70+45+42+35+18=630
99 2,8,9,10,12,18,40 LCM 360 180+45+40+36+30+20+9=360
99 3,4,5,6,36,45 LCM 180 60+45+36+30+5+4=180
99 3,4,6,10,12,24,40 LCM 120 40+30+20+12+10+5+3=120
99 3,4,6,8,18,24,36 LCM 72 24+18+12+9+4+3+2=72
99 3,5,8,9,12,18,20,24 LCM 360 120+72+45+40+30+20+18+15=360
99 4,5,6,8,10,12,24,30 LCM 120 30+24+20+15+12+10+5+4=120
100 2,6,7,8,21,56 LCM 168 84+28+24+21+8+3=168
100 3,4,7,8,12,24,42 LCM 168 56+42+24+21+14+7+4=168
100 3,5,6,8,9,24,45 LCM 360 120+72+60+45+40+15+8=360
101 2,4,5,30,60 LCM 60 30+15+12+2+1=60
101 2,5,10,15,20,21,28 LCM 420 210+84+42+28+21+20+15=420
101 2,6,8,12,21,24,28 LCM 168 84+28+21+14+8+7+6=168
101 2,6,9,11,18,22,33 LCM 198 99+33+22+18+11+9+6=198
101 3,4,5,10,21,28,30 LCM 420 140+105+84+42+20+15+14=420
102 2,4,8,16,24,48 LCM 48 24+12+6+3+2+1=48
102 2,4,9,12,30,45 LCM 180 90+45+20+15+6+4=180
102 2,6,8,12,20,24,30 LCM 120 60+20+15+10+6+5+4=120
102 3,4,6,10,15,16,48 LCM 240 80+60+40+24+16+15+5=240
102 3,4,6,8,9,72 LCM 72 24+18+12+9+8+1=72
102 3,4,6,9,15,20,45 LCM 180 60+45+30+20+12+9+4=180
102 3,5,7,10,14,15,20,28 LCM 420 140+84+60+42+30+28+21+15=420
102 3,6,7,8,12,14,24,28 LCM 168 56+28+24+21+14+12+7+6=168
103 2,4,8,14,35,40 LCM 280 140+70+35+20+8+7=280
103 2,5,12,14,15,20,35 LCM 420 210+84+35+30+28+21+12=420
103 2,5,9,18,20,21,28 LCM 1260 630+252+140+70+63+60+45=1260
103 2,7,9,10,15,18,42 LCM 630 315+90+70+63+42+35+15=630
103 3,4,5,12,14,30,35 LCM 420 140+105+84+35+30+14+12=420
103 3,4,5,12,15,24,40 LCM 120 40+30+24+10+8+5+3=120
103 3,5,6,7,10,30,42 LCM 210 70+42+35+30+21+7+5=210
104 2,3,20,21,28,30 LCM 420 210+140+21+20+15+14=420
104 2,4,7,21,28,42 LCM 84 42+21+12+4+3+2=84
104 2,6,8,14,15,24,35 LCM 840 420+140+105+60+56+35+24=840
104 3,4,6,9,16,18,48 LCM 144 48+36+24+16+9+8+3=144
104 3,5,7,9,14,18,20,28 LCM 1260 420+252+180+140+90+70+63+45=1260
104 4,5,6,7,10,14,28,30 LCM 420 105+84+70+60+42+30+15+14=420
105 2,4,7,20,30,42 LCM 420 210+105+60+21+14+10=420
105 2,5,11,12,20,22,33 LCM 660 330+132+60+55+33+30+20=660
105 4,5,6,8,10,12,20,40 LCM 120 30+24+20+15+12+10+6+3=120
105 4,5,6,8,9,15,18,40 LCM 360 90+72+60+45+40+24+20+9=360
106 2,5,10,15,18,20,36 LCM 180 90+36+18+12+10+9+5=180
106 2,5,6,12,36,45 LCM 180 90+36+30+15+5+4=180
106 2,6,8,11,22,24,33 LCM 264 132+44+33+24+12+11+8=264
106 2,6,8,12,18,24,36 LCM 72 36+12+9+6+4+3+2=72
106 3,4,5,10,18,30,36 LCM 180 60+45+36+18+10+6+5=180
106 3,4,5,10,20,24,40 LCM 120 40+30+24+12+6+5+3=120
106 3,4,6,8,15,30,40 LCM 120 40+30+20+15+8+4+3=120
106 3,6,8,9,10,12,18,40 LCM 360 120+60+45+40+36+30+20+9=360
106 4,5,6,9,10,12,15,45 LCM 180 45+36+30+20+18+15+12+4=180
106 4,6,8,9,10,12,15,18,24 LCM 360 90+60+45+40+36+30+24+20+15=360
Greg Egan has verified, using Mathematica, that $77$ cannot be written as the sum of distinct positive integers whose reciprocals sum to $1$:
https://twitter.com/gregeganSF/status/1156839799610675200.
Christopher D. Long proved it using Sage; his code is here:
https://github.com/octonion/puzzles/tree/master/twitter/lehmer.
Dima Pasechnik wrote a very elegant Sage program that proves this result:
sage: from builtins import sum as psum
sage: def sumreseq1(l):
....: return 1==psum(map(lambda t: 1/t, l))
sage: filter(sumreseq1, Partitions(11,max_slope=-1))
[[6, 3, 2]]
sage: filter(sumreseq1, Partitions(77,max_slope=-1))
[]
There are $58499$ ways to write $77$ as a sum of distinct positive integers, and one only needs to check these to prove the result.
Here, by the way, is the complete list of positive integers that cannot be written as the sum of distinct positive integers whose reciprocals sum to 1:
$2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 33, 34, 35, 36, 39, 40, 41, 42, 44, 46, 47, 48, 49, 51, 56, 58, 63, 68, 70, 72, 77$
This is $A051882$ in the Online Encyclopedia of Integer Sequences. However, I don't believe there is a reference to a proof. Luckily, Mr. Drake has written a program that verifies this list.
Here's an attempt at a human-readable proof. Let such a partition be $X = \{ x_i \}, 1 \leq i \leq n$. Note that by modular considerations there cannot be a prime $p$ that only divides a single $x_i$. In particular, this implies that the largest such prime must satisfy $p + 2\cdot p <= 77$, and hence $p \leq 23$. But more is true; if $ Y = \{ p\cdot y_i \}, 1 \leq i \leq m$, are the terms of $X$ divisible by $p$, then $ \sum_{i=1}^{m} y/y_i \equiv 0 \pmod p$, where $ y = \prod_{i=1}^{m} y_i$. This rapidly eliminates $p > 11$.
Now $p=11$ is only slightly harder; the only candidate for $Y$ is $\{ 1\cdot 11,2\cdot 11,3\cdot 11 \}$, but this only leaves $77-6\cdot 11 = 11$ for the sum of the remaining terms of our partition. Now, the remaining terms can only be products of $2$ and $3$, leaving the only candidate for the remaining terms $\{ 2,3,6 \}$, but $1/2+1/3+1/6 = 1$ is too big. Thus the largest possible prime factor dividing an $x_i$ is 7.
The above argument also works for prime powers, showing that $3^2$ is the largest squared prime factor that may occur and any higher power must be a power of $2$. With some more work, you get a list of candidate $Y$ if $7$ divides some $x_i$ e.g. $Y = \{ 7, 42 \}, \{ 14, 35 \}, \{ 21, 28 \}$ or $Y = \{ 7, 14, 28\}$.
This should reduce the possible $X$ to a list checkable by hand.