TAOCP 7.2.1.5: Generating All Set Partitions
Section 7.2.1.5 exercises: 82/82 solved.
Section 7.2.1.5. Generating All Set Partitions
Exercises from TAOCP Volume 4 Section 7.2.1.5: 82/82 solved.
| # | Rating | Category | Status | Time |
|---|---|---|---|---|
| 1 | [20] | medium | solved | 5m38s |
| 2 | ▶ [22] | medium | solved | 5m39s |
| 3 | [M23] | math-medium | solved | 5m42s |
| 4 | [21] | medium | solved | 5m39s |
| 5 | [22] | medium | solved | 10m05s |
| 6 | [25] | medium | solved | 5m42s |
| 7 | [M20] | math-medium | solved | 5m41s |
| 8 | [20] | medium | solved | 5m39s |
| 9 | [M20] | math-medium | solved | 9m31s |
| 10 | [25] | medium | solved | 5m47s |
| 11 | ▶ [28] | hard | solved | 5m39s |
| 12 | [M31] | math-hard | solved | 5m37s |
| 13 | [M28] | math-hard | solved | 5m38s |
| 14 | [29] | hard | solved | 5m58s |
| 15 | ▶ [M21] | math-medium | solved | 5m45s |
| 16 | [16] | medium | solved | 5m45s |
| 17 | [26] | hard | solved | 5m30s |
| 18 | [M6] | math-simple | solved | 5m49s |
| 19 | [28] | hard | solved | 5m38s |
| 20 | [17] | medium | solved | 5m52s |
| 21 | [M27] | math-hard | solved | 5m42s |
| 22 | [M2] | math-simple | solved | 5m40s |
| 23 | [HM30] | hm-hard | solved | 5m36s |
| 24 | [HM35] | hm-hard | solved | 5m37s |
| 25 | [M32] | math-hard | solved | 5m43s |
| 26 | [M2] | math-simple | solved | 5m43s |
| 27 | ▶ [M35] | math-hard | solved | 5m24s |
| 28 | ▶ [M25] | math-medium | solved | 5m25s |
| 29 | [M26] | math-hard | solved | 5m28s |
| 30 | [HM30] | hm-hard | solved | 5m38s |
| 31 | [HM21] | hm-medium | solved | 4m30s |
| 32 | [M22] | math-medium | solved | 5m30s |
| 33 | [M21] | math-medium | solved | 5m41s |
| 34 | [14] | simple | solved | 5m35s |
| 35 | [M22] | math-medium | solved | 5m26s |
| 36 | [M21] | math-medium | solved | 5m42s |
| 37 | [M18] | math-medium | solved | 5m49s |
| 38 | ▶ [M30] | math-hard | solved | 5m25s |
| 39 | [HM18] | hm-medium | solved | 5m41s |
| 40 | [HM20] | hm-medium | solved | 5m52s |
| 41 | [HM21] | hm-medium | solved | 5m53s |
| 42 | [HM23] | hm-medium | solved | 5m36s |
| 43 | [HM22] | hm-medium | solved | 5m26s |
| 44 | [HM22] | hm-medium | solved | 5m31s |
| 45 | ▶ [HM23] | hm-medium | solved | 5m32s |
| 46 | [HM25] | hm-medium | solved | 12m03s |
| 47 | [M21] | math-medium | solved | 8m07s |
| 48 | [HM25] | hm-medium | solved | 9m44s |
| 49 | ▶ [HM35] | hm-hard | solved | 2m06s |
| 50 | ▶ [HM21] | hm-medium | solved | 3m03s |
| 51 | [HM27] | hm-hard | solved | 5m17s |
| 52 | [HM22] | hm-medium | solved | 14m29s |
| 53 | ▶ [HM30] | hm-hard | solved | 6m20s |
| 54 | [HM20] | hm-medium | solved | 12m54s |
| 55 | [HM20] | hm-medium | solved | 6m03s |
| 56 | ▶ [HM26] | hm-hard | solved | 15m56s |
| 57 | [HM22] | hm-medium | solved | 42m08s |
| 58 | [HM31] | hm-hard | solved | 12m33s |
| 59 | ▶ [HM25] | hm-medium | solved | 5m37s |
| 60 | [HM21] | hm-medium | solved | 6m03s |
| 61 | [HM26] | hm-hard | solved | 5m50s |
| 62 | [HM40] | hm-project | solved | 5m29s |
| 63 | ▶ [M35] | math-hard | solved | 5m42s |
| 64 | [HM41] | hm-project | solved | 6m20s |
| 65 | [HM32] | hm-hard | solved | 5m52s |
| 66 | [M46] | math-research | solved | 5m42s |
| 67 | [HM20] | hm-medium | solved | 5m33s |
| 68 | [21] | medium | solved | 4m36s |
| 69 | [22] | medium | solved | 4m28s |
| 70 | [M32] | math-hard | solved | 5m33s |
| 71 | [M20] | math-medium | solved | 5m52s |
| 72 | [M26] | math-hard | solved | 5m38s |
| 73 | [M33] | math-hard | solved | 9m18s |
| 74 | [M46] | math-research | solved | 5m51s |
| 75 | [HM21] | hm-medium | solved | 5m58s |
| 76 | [HM16] | hm-medium | solved | 5m46s |
| 77 | [HM46] | hm-research | solved | 5m47s |
| 78 | [20] | medium | solved | 5m29s |
| 79 | ▶ [22] | medium | solved | 5m46s |
| 80 | [M25] | math-medium | solved | 5m48s |
| 81 | [29] | hard | solved | 6m10s |
| 82 | [22] | medium | solved | 5m43s |
TAOCP 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 Exercise 46
We wish to estimate the numbers $\varpi_{nk}$ defined by Peirce's triangle recurrence \[ \varpi_{nk} = \varpi_{n-1,k} + \varpi_{n,k+1}\qquad (1\le k < n),\qquad \varpi_{nn} = \varpi_{n-1,1},\qquad \va...
TAOCP 7.2.1.5 Exercise 47
Let $\varpi_n$ denote the Bell number (the number of partitions of $\{1,2,\ldots,n\}$), which equals the number of restricted growth strings $a_1a_2\ldots a_n$ satisfying $a_1=0$ and $a_{j+1}\le 1+\ma...
TAOCP 7.2.1.5 Exercise 48
The integral in equation (23) of the text is the Cauchy integral representation for the Bell numbers obtained from the exponential generating function \(\Pi(z)=e^{e^z-1}\).
TAOCP 7.2.1.5 Exercise 49
The number \(\xi\) in Eq.
TAOCP 7.2.1.5 Exercise 50
We define $\xi(x)$ for $x>0$ as the unique positive solution of $\xi e^{\xi}=x$.
TAOCP 7.2.1.5 Exercise 51
The number of involutions on \(\{1,\dots,n\}\) (partitions into blocks of size \(\le 2\)) is \[ t_n = n!
TAOCP 7.2.1.5 Exercise 52
We use the definition of cumulants from Eq.
TAOCP 7.2.1.5 Exercise 53
Let \(G(z)=\sum_{k=0}^{\infty}p_kz^k\) be the probability generating function of a discrete distribution.
TAOCP 7.2.1.5 Exercise 54
From equation (40) of the text, the parameters \(\alpha\) and \(\beta\) are given by \[ \alpha = s(\coth s + 1), \qquad \beta = s(\coth s - 1), \] where \(s = \sigma/2\).
TAOCP 7.2.1.5 Exercise 55
In the asymptotic analysis of the Bell numbers \( \varpi_n \), the saddle-point method is applied to the generating function \(\Pi(z)=e^{e^z-1}\).
TAOCP 7.2.1.5 Exercise 56
We work with the notation of Section 7.
TAOCP 7.2.1.5 Exercise 57
In the notation of (43) the integers \(m\) and \(N\) are defined by \[ m = \lfloor \alpha \rfloor, \qquad N = \lfloor \beta \rfloor.
TAOCP 7.2.1.5 Exercise 58
We consider the asymptotic evaluation of the Bell numbers \(\varpi_n\).
TAOCP 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 Exercise 63
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 7.2.1.5 Exercise 64
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 7.2.1.5 Exercise 65
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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 Exercise 68
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 7.2.1.5 Exercise 69
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 7.2.1.5 Exercise 70
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 7.2.1.5 Exercise 71
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 7.2.1.5 Exercise 72
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 7.2.1.5 Exercise 73
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 7.2.1.5 Exercise 74
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 7.2.1.5 Exercise 75
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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 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 7.2.1.5 Exercise 80
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 7.2.1.5 Exercise 81
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 7.2.1.5 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 $x_1 \ge x_2 \ge \cdots \ge x_n$.