As the table below shows, (134, 135, 136) is the largest triplet that works, and 134 + 135 + 136 = 405.
x y z sum of reciprocal
reciprocals of sum
1 2 3 11/6 0.5454545
2 3 4 13/12 0.9230769
3 4 5 47/60 1.2765957
4 5 6 37/60 1.6216216
5 6 7 107/210 1.9626168
6 7 8 73/168 2.3013699
7 8 9 191/504 2.6387435
8 9 10 121/360 2.9752066
9 10 11 299/990 3.3110368
10 11 12 181/660 3.6464088
11 12 13 431/1716 3.9814385
12 13 14 253/1092 4.3162055
13 14 15 587/2730 4.6507666
14 15 16 337/1680 4.9851632
15 16 17 767/4080 5.3194263
16 17 18 433/2448 5.6535797
17 18 19 971/5814 5.9876416
18 19 20 541/3420 6.3216266
19 20 21 1199/7980 6.6555463
20 21 22 661/4620 6.9894100
21 22 23 1451/10626 7.3232254
22 23 24 793/6072 7.6569987
23 24 25 1727/13800 7.9907354
24 25 26 937/7800 8.3244397
25 26 27 2027/17550 8.6581154
26 27 28 1093/9828 8.9917658
27 28 29 2351/21924 9.3253934
28 29 30 1261/12180 9.6590008
29 30 31 2699/26970 9.9925898
30 31 32 1441/14880 10.3261624
31 32 33 3071/32736 10.6597200
32 33 34 1633/17952 10.9932639
33 34 35 3467/39270 11.3267955
34 35 36 1837/21420 11.6603157
35 36 37 3887/46620 11.9938256
36 37 38 2053/25308 12.3273259
37 38 39 4331/54834 12.6608174
38 39 40 2281/29640 12.9943007
39 40 41 4799/63960 13.3277766
40 41 42 2521/34440 13.6612455
41 42 43 5291/74046 13.9947080
42 43 44 2773/39732 14.3281644
43 44 45 5807/85140 14.6616153
44 45 46 3037/45540 14.9950609
45 46 47 6347/97290 15.3285017
46 47 48 3313/51888 15.6619378
47 48 49 6911/110544 15.9953697
48 49 50 3601/58800 16.3287976
49 50 51 7499/124950 16.6622216
50 51 52 3901/66300 16.9956421
51 52 53 8111/140556 17.3290593
52 53 54 4213/74412 17.6624733
53 54 55 8747/157410 17.9958843
54 55 56 4537/83160 18.3292925
55 56 57 9407/175560 18.6626980
56 57 58 4873/92568 18.9961010
57 58 59 10091/195054 19.3295015
58 59 60 5221/102660 19.6628998
59 60 61 10799/215940 19.9962960
60 61 62 5581/113460 20.3296900
61 62 63 11531/238266 20.6630821
62 63 64 5953/124992 20.9964724
63 64 65 12287/262080 21.3298608
64 65 66 6337/137280 21.6632476
65 66 67 13067/287430 21.9966327
66 67 68 6733/150348 22.3300163
67 68 69 13871/314364 22.6633985
68 69 70 7141/164220 22.9967792
69 70 71 14699/342930 23.3301585
70 71 72 7561/178920 23.6635366
71 72 73 15551/373176 23.9969134
72 73 74 7993/194472 24.3302890
73 74 75 16427/405150 24.6636635
74 75 76 8437/210900 24.9970369
75 76 77 17327/438900 25.3304092
76 77 78 8893/228228 25.6637805
77 78 79 18251/474474 25.9971508
78 79 80 9361/246480 26.3305202
79 80 81 19199/511920 26.6638887
80 81 82 9841/265680 26.9972564
81 82 83 20171/551286 27.3306232
82 83 84 10333/285852 27.6639892
83 84 85 21167/592620 27.9973544
84 85 86 10837/307020 28.3307188
85 86 87 22187/635970 28.6640826
86 87 88 11353/329208 28.9974456
87 88 89 23231/681384 29.3308080
88 89 90 11881/352440 29.6641697
89 90 91 24299/728910 29.9975308
90 91 92 12421/376740 30.3308912
91 92 93 25391/778596 30.6642511
92 93 94 12973/402132 30.9976104
93 94 95 26507/830490 31.3309692
94 95 96 13537/428640 31.6643274
95 96 97 27647/884640 31.9976851
96 97 98 14113/456288 32.3310423
97 98 99 28811/941094 32.6643990
98 99 100 14701/485100 32.9977553
99 100 101 29999/999900 33.3311110
100 101 102 15301/515100 33.6644664
101 102 103 31211/1061106 33.9978213
102 103 104 15913/546312 34.3311758
103 104 105 32447/1124760 34.6645298
104 105 106 16537/578760 34.9978835
105 106 107 33707/1190910 35.3312368
106 107 108 17173/612468 35.6645898
107 108 109 34991/1259604 35.9979423
108 109 110 17821/647460 36.3312945
109 110 111 36299/1330890 36.6646464
110 111 112 18481/683760 36.9979979
111 112 113 37631/1404816 37.3313492
112 113 114 19153/721392 37.6647000
113 114 115 38987/1481430 37.9980506
114 115 116 19837/760380 38.3314009
115 116 117 40367/1560780 38.6647509
116 117 118 20533/800748 38.9981006
117 118 119 41771/1642914 39.3314500
118 119 120 21241/842520 39.6647992
119 120 121 43199/1727880 39.9981481
120 121 122 21961/885720 40.3314967
121 122 123 44651/1815726 40.6648451
122 123 124 22693/930372 40.9981933
123 124 125 46127/1906500 41.3315412
124 125 126 23437/976500 41.6648889
125 126 127 47627/2000250 41.9982363
126 127 128 24193/1024128 42.3315835
127 128 129 49151/2097024 42.6649305
128 129 130 24961/1073280 42.9982773
129 130 131 50699/2196870 43.3316239
130 131 132 25741/1123980 43.6649703
131 132 133 52271/2299836 43.9983165
132 133 134 26533/1176252 44.3316625
133 134 135 53867/2405970 44.6650083
134 135 136 27337/1230120 44.9983539
135 136 137 55487/2515320 45.3316993
clc,clearvars
for n=sym(1):1000
v=1/n+1/(n+1)+1/(n+2);
fprintf('%5s %4s %4s %-15s %9.7f\n',n,n+1,n+2,v,double(1/v))
if v<=sym(1/45)
break
end
end
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Posted by Charlie
on 2023-07-24 08:15:57 |
Solution (spoiler)
