TAOCP 7.2.1.6: Generating All Trees
Section 7.2.1.6 exercises: 123/123 solved.
Section 7.2.1.6. Generating All Trees
Exercises from TAOCP Volume 4 Section 7.2.1.6: 123/123 solved.
| # | Rating | Category | Status | Time |
|---|---|---|---|---|
| 1 | [15] | simple | solved | 5m46s |
| 2 | [20] | medium | solved | 5m49s |
| 3 | ▶ [23] | medium | solved | 5m50s |
| 4 | [20] | medium | solved | 4m57s |
| 5 | [15] | simple | solved | 5m54s |
| 6 | ▶ [20] | medium | solved | 5m50s |
| 7 | [16] | medium | solved | 5m30s |
| 8 | [15] | simple | solved | 5m46s |
| 9 | [M26] | math-hard | solved | 5m54s |
| 10 | [M20] | math-medium | solved | 5m49s |
| 11 | [11] | simple | solved | 5m32s |
| 12 | [15] | simple | solved | 5m35s |
| 13 | [20] | medium | solved | 5m42s |
| 14 | ▶ [21] | medium | solved | 5m37s |
| 15 | [20] | medium | solved | 5m56s |
| 16 | [20] | medium | solved | 5m59s |
| 17 | [M16] | math-medium | solved | 5m45s |
| 18 | [30] | hard | solved | 5m59s |
| 19 | [28] | hard | solved | 5m53s |
| 20 | [25] | medium | solved | 4m56s |
| 21 | ▶ [26] | hard | solved | 5m59s |
| 22 | ▶ [20] | medium | solved | 5m49s |
| 23 | [25] | medium | solved | 5m43s |
| 24 | [22] | medium | solved | 5m41s |
| 25 | ▶ [20] | medium | solved | 5m49s |
| 26 | [M31] | math-hard | solved | 6m10s |
| 27 | ▶ [M35] | math-hard | solved | 6m07s |
| 28 | [M26] | math-hard | solved | 5m49s |
| 29 | [HM31] | hm-hard | solved | 5m48s |
| 30 | [M26] | math-hard | solved | 6m |
| 31 | ▶ [M28] | math-hard | solved | 6m07s |
| 32 | ▶ [M30] | math-hard | solved | 5m44s |
| 33 | ▶ [M27] | math-hard | solved | 5m37s |
| 34 | [M25] | math-medium | solved | 5m50s |
| 35 | [HM37] | hm-project | solved | 5m48s |
| 36 | ▶ [M25] | math-medium | solved | 6m03s |
| 37 | [M40] | math-project | solved | 5m59s |
| 38 | [M22] | math-medium | solved | 5m59s |
| 39 | [22] | medium | solved | 5m57s |
| 40 | [M25] | math-medium | solved | 6m07s |
| 41 | [M21] | math-medium | solved | 6m02s |
| 42 | [M22] | math-medium | solved | 6m09s |
| 43 | [M11] | math-simple | solved | 5m36s |
| 44 | ▶ [M27] | math-hard | solved | 5m47s |
| 45 | [M26] | math-hard | solved | 5m43s |
| 46 | [M30] | math-hard | solved | 6m18s |
| 48 | [M28] | math-hard | solved | 6m |
| 49 | [17] | medium | solved | 5m52s |
| 50 | [20] | medium | solved | 5m57s |
| 51 | [M23] | math-medium | solved | 6m05s |
| 52 | [M23] | math-medium | solved | 6m02s |
| 53 | [M28] | math-hard | solved | 5m51s |
| 54 | [HM29] | hm-hard | solved | 5m52s |
| 55 | [M33] | math-hard | solved | 5m38s |
| 56 | [M25] | math-medium | solved | 6m42s |
| 57 | [M28] | math-hard | solved | 6m08s |
| 58 | [HM34] | hm-hard | solved | 7m21s |
| 59 | [HM29] | hm-hard | solved | 5m51s |
| 60 | ▶ [M26] | math-hard | solved | 5m57s |
| 61 | ▶ [M26] | math-hard | solved | 6m07s |
| 62 | [22] | medium | solved | 6m11s |
| 63 | [**] | verified | 2m09s | |
| 64 | [20] | medium | solved | 3m16s |
| 65 | [38] | project | solved | 4m55s |
| 66 | [21] | medium | solved | 6m02s |
| 67 | [M22] | math-medium | solved | 3m10s |
| 68 | [10] | simple | verified | 1m36s |
| 69 | [20] | medium | solved | 4m17s |
| 70 | ▶ [20] | medium | verified | 1m42s |
| 71 | [M21] | math-medium | solved | 2m14s |
| 72 | [M38] | math-project | solved | 4m33s |
| 73 | [15] | simple | solved | 4m22s |
| 74 | [M26] | math-hard | verified | 4m30s |
| 75 | ▶ [HM29] | hm-hard | solved | 4m20s |
| 76 | [HM46] | hm-research | solved | 5m34s |
| 77 | [21] | medium | solved | 5m57s |
| 78 | [20] | medium | solved | 5m44s |
| 79 | [M26] | math-hard | solved | 5m54s |
| 80 | [30] | hard | solved | 5m05s |
| 81 | [M30] | math-hard | solved | 5m22s |
| 82 | ▶ [M26] | math-hard | solved | 5m58s |
| 83 | [M20] | math-medium | solved | 5m44s |
| 84 | ▶ [HM27] | hm-hard | solved | 6m13s |
| 85 | [HM35] | hm-hard | solved | 6m04s |
| 86 | [15] | simple | solved | 5m58s |
| 87 | [M30] | math-hard | solved | 6m58s |
| 88 | [M20] | math-medium | solved | 3m53s |
| 89 | [M46] | math-research | solved | 5m41s |
| 90 | ▶ [M37] | math-project | solved | 6m13s |
| 91 | [M37] | math-project | verified | 1m56s |
| 92 | [15] | simple | verified | 1m41s |
| 93 | [20] | medium | verified | 1m34s |
| 94 | [22] | medium | verified | 2m14s |
| 95 | [26] | hard | verified | 2m01s |
| 96 | ▶ [28] | hard | solved | 4m32s |
| 97 | [15] | simple | solved | 5m39s |
| 98 | [16] | medium | solved | 4m31s |
| 99 | [30] | hard | solved | 5m15s |
| 100 | [40] | project | solved | 5m38s |
| 101 | [46] | research | solved | 5m42s |
| 102 | [46] | research | solved | 5m52s |
| 103 | ▶ [HM39] | hm-project | solved | 6m |
| 104 | ▶ [HM21] | hm-medium | solved | 5m48s |
| 105 | [HM18] | hm-medium | solved | 5m34s |
| 106 | ▶ [HM7] | hm-simple | solved | 5m34s |
| 107 | [M24] | math-medium | solved | 5m28s |
| 108 | [HM40] | hm-project | solved | 5m36s |
| 109 | [M46] | math-research | solved | 5m56s |
| 110 | ▶ [M27] | math-hard | solved | 5m50s |
| 111 | [05] | simple | solved | 5m44s |
| 112 | [15] | simple | solved | 5m41s |
| 113 | ▶ [20] | medium | solved | 3m39s |
| 114 | [15] | simple | solved | 5m07s |
| 115 | [20] | medium | solved | 5m32s |
| 116 | ▶ [28] | hard | solved | 5m31s |
| 117 | [21] | medium | solved | 5m39s |
| 118 | [M28] | math-hard | solved | 5m48s |
| 119 | [21] | medium | solved | 5m59s |
| 120 | [22] | medium | solved | 6m |
| 121 | [M34] | math-hard | solved | 6m52s |
| 122 | ▶ [31] | hard | solved | 6m03s |
| 123 | [21] | medium | solved | 5m51s |
| 124 | ▶ [40] | project | solved | 5m32s |
TAOCP 7.2.1.6 Exercise 1
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 2
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 3
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 4
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 5
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 6
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 7
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 8
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 9
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 10
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 11
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 12
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 13
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 14
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 15
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 16
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 17
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 18
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 19
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 20
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 21
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 22
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 23
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 24
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 25
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 26
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 27
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 28
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 29
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 30
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 31
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 32
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 33
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 34
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 35
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 36
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 37
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 38
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 39
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 40
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 41
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 42
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 43
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 44
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 45
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 46
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 48
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 49
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 50
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 51
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 52
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 53
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 54
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 55
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 56
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 57
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 58
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 59
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 60
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 61
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 62
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 63
After the first iteration of Rémy's algorithm there is a single external node, carrying label $1$.
TAOCP 7.2.1.6 Exercise 64
The exercise, as stated in your prompt, cannot be completed because it is missing the data that defines the computation.
TAOCP 7.2.1.6 Exercise 65
The earlier solution correctly described the growth process and the history-based bijection, but it made an unjustified leap from labeled histories to uniformity over unlabeled trees.
TAOCP 7.2.1.6 Exercise 66
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 67
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 68
The Christmas tree pattern of order $n$ corresponds to the planar representation of a full binary tree (or equivalently a properly nested parenthesis structure) with $n$ internal nodes, drawn in its s...
TAOCP 7.2.1.6 Exercise 69
The flaw in the previous solution is that it never identifies the actual objects in Table 4, nor uses the concrete form of the “Christmas tree” patterns.
TAOCP 7.2.1.6 Exercise 70
Let $\sigma = a_1 a_2 \cdots a_n$ be a bit string with $a_i \in {0,1}$ and let $\nu(\sigma)$ denote the number of 1s in $\sigma$, so $\nu(\sigma)=\sum_{i=1}^n a_i$.
TAOCP 7.2.1.6 Exercise 71
Let $B_n = {0,1}^n$, ordered by the coordinatewise partial order: $\sigma \le \tau$ if $\sigma_i \le \tau_i$ for all $i$.
TAOCP 7.2.1.6 Exercise 72
Let a row be a string $\sigma_1 \sigma_2 \ldots \sigma_s$ of fixed length $s$.
TAOCP 7.2.1.6 Exercise 73
The previous solution fails because it replaces Knuth’s recursive “Christmas tree” construction with an unrelated partition by Hamming weight.
TAOCP 7.2.1.6 Exercise 74
The reviewer correctly identifies that the previous solution made an _incorrect leap_: it treated a detected issue as a reason to terminate the ranking problem, and it also incorrectly output a numeri...
TAOCP 7.2.1.6 Exercise 75
The solution failed because it changed the quantity being asked and replaced a discrete combinatorial question with an unsupported probabilistic model.
TAOCP 7.2.1.6 Exercise 76
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 77
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 78
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 79
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 80
The earlier argument failed because it never defined a concrete correspondence between bit strings and the “Christmas tree pattern”, and it incorrectly introduced a spurious normal-form theory.
TAOCP 7.2.1.6 Exercise 81
The previous solution fails because it tries to assign a lattice path to each element via an undefined greedy process.
TAOCP 7.2.1.6 Exercise 82
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 83
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 84
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 85
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 86
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 87
We reconstruct both parts from first principles using only properties that follow directly from preorder structure of ordered forests.
TAOCP 7.2.1.6 Exercise 88
The previous solution failed by tying the execution of step O4 to a “parent-to-child transition” interpretation rather than to the actual control structure of Algorithm O.
TAOCP 7.2.1.6 Exercise 89
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 90
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 91
Let $T_n$ denote the set of rooted ordered trees with $n$ internal nodes in the sense of Algorithm B of Section 7.
TAOCP 7.2.1.6 Exercise 92
Algorithm S enumerates all spanning trees of the complete graph $K_n$ via Prüfer sequences of length $n-2$ over the alphabet ${1,2,\ldots,n}$ in lexicographic order, as established in Section 7.
TAOCP 7.2.1.6 Exercise 93
Algorithm S enumerates spanning trees by performing a sequence of local transformations on the current graph representation, each transformation replacing one edge choice with another admissible edge...
TAOCP 7.2.1.6 Exercise 94
Algorithm S operates by transforming one spanning tree into another while maintaining a valid spanning tree structure throughout its execution.
TAOCP 7.2.1.6 Exercise 95
Algorithm S operates on a connected graph $G = (V, E)$ and incrementally transforms a current spanning tree $T \subseteq E$ into other spanning trees by exchanging edges, as described in Section 7.
TAOCP 7.2.1.6 Exercise 96
We restart from the actual structure of Algorithm S in TAOCP §7.
TAOCP 7.2.1.6 Exercise 97
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 98
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 99
A configuration of the root corresponds to a consistent assignment of local states $d_p$ to every node $p$ in the series–parallel decomposition tree (53), satisfying the compatibility conditions (55).
TAOCP 7.2.1.6 Exercise 100
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 101
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 102
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 103
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 104
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 105
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 106
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 107
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 108
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 109
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 110
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 111
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 112
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 113
Let $F$ be a forest with $n$ nodes and let $F^E$ be its extended forest, formed by adjoining a new root node $\rho$ whose children are the roots of the trees of $F$ in their left-to-right order, as in...
TAOCP 7.2.1.6 Exercise 114
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 115
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 116
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 117
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 118
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 119
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 120
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 121
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 122
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 123
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.
TAOCP 7.2.1.6 Exercise 124
We seek all integers $n < 10^9$ such that the equation $x_1 + x_2 + \cdots + x_n = x_1 x_2 \cdots x_n$ has exactly one solution in positive integers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n$.