## Puzzle #2 – Sum of aliquot divisors of a and b

Find pair of integers (a, b) with the property that the sum of the aliquot divisors of a exceeds the sum of the aliquot divisors of by a square number.

Here are two examples:

99   has 6 divisors:     1,   3,   9,   11,   33,   99
Sum of all divisors:     156
The sum of its aliquot divisors:     156 – 99 = 57

63   has 6 divisors:     1,   3,   7,   9,   21,   63
Sum of all divisors:     104
The sum of its aliquot divisors:     104 – 63 = 41

and,   $(156 - 99) - (104 - 63) = 16 = 4^2$   a square number

similarly for   325   and   175 :

175   has 6 divisors: 1,   5,   7,   25,   35,   175
Sum of all divisors:     248
The sum of its aliquot divisors:     248 – 175 = 73

325   has 6 divisors: 1,   5,   13,   25,   65,   325
Sum of all divisors:     434
The sum of its aliquot divisors:     434 – 325 = 109

and,   $109 - 73 = 36 = 6^2$

Find other pairs

math grad - Interest: Number theory
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### 5 Responses to Puzzle #2 – Sum of aliquot divisors of a and b

1. paul says:

There seems to be a lot of these, here are just those with a <= 40. at a = 100 there are 294 solutions.

Sum of the aliquot divisors of 14 = 10
Sum of the aliquot divisors of 2 = 1
and 10 – 1 = 9 = 3^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 2 = 1
and 17 – 1 = 16 = 4^2

Sum of the aliquot divisors of 40 = 50
Sum of the aliquot divisors of 2 = 1
and 50 – 1 = 49 = 7^2

Sum of the aliquot divisors of 14 = 10
Sum of the aliquot divisors of 3 = 1
and 10 – 1 = 9 = 3^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 3 = 1
and 17 – 1 = 16 = 4^2

Sum of the aliquot divisors of 40 = 50
Sum of the aliquot divisors of 3 = 1
and 50 – 1 = 49 = 7^2

Sum of the aliquot divisors of 8 = 7
Sum of the aliquot divisors of 4 = 3
and 7 – 3 = 4 = 2^2

Sum of the aliquot divisors of 28 = 28
Sum of the aliquot divisors of 4 = 3
and 28 – 3 = 25 = 5^2

Sum of the aliquot divisors of 14 = 10
Sum of the aliquot divisors of 5 = 1
and 10 – 1 = 9 = 3^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 5 = 1
and 17 – 1 = 16 = 4^2

Sum of the aliquot divisors of 40 = 50
Sum of the aliquot divisors of 5 = 1
and 50 – 1 = 49 = 7^2

Sum of the aliquot divisors of 14 = 10
Sum of the aliquot divisors of 6 = 6
and 10 – 6 = 4 = 2^2

Sum of the aliquot divisors of 16 = 15
Sum of the aliquot divisors of 6 = 6
and 15 – 6 = 9 = 3^2

Sum of the aliquot divisors of 20 = 22
Sum of the aliquot divisors of 6 = 6
and 22 – 6 = 16 = 4^2

Sum of the aliquot divisors of 30 = 42
Sum of the aliquot divisors of 6 = 6
and 42 – 6 = 36 = 6^2

Sum of the aliquot divisors of 32 = 31
Sum of the aliquot divisors of 6 = 6
and 31 – 6 = 25 = 5^2

Sum of the aliquot divisors of 33 = 15
Sum of the aliquot divisors of 6 = 6
and 15 – 6 = 9 = 3^2

Sum of the aliquot divisors of 36 = 55
Sum of the aliquot divisors of 6 = 6
and 55 – 6 = 49 = 7^2

Sum of the aliquot divisors of 38 = 22
Sum of the aliquot divisors of 6 = 6
and 22 – 6 = 16 = 4^2

Sum of the aliquot divisors of 14 = 10
Sum of the aliquot divisors of 7 = 1
and 10 – 1 = 9 = 3^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 7 = 1
and 17 – 1 = 16 = 4^2

Sum of the aliquot divisors of 40 = 50
Sum of the aliquot divisors of 7 = 1
and 50 – 1 = 49 = 7^2

Sum of the aliquot divisors of 12 = 16
Sum of the aliquot divisors of 8 = 7
and 16 – 7 = 9 = 3^2

Sum of the aliquot divisors of 21 = 11
Sum of the aliquot divisors of 8 = 7
and 11 – 7 = 4 = 2^2

Sum of the aliquot divisors of 26 = 16
Sum of the aliquot divisors of 8 = 7
and 16 – 7 = 9 = 3^2

Sum of the aliquot divisors of 10 = 8
Sum of the aliquot divisors of 9 = 4
and 8 – 4 = 4 = 2^2

Sum of the aliquot divisors of 27 = 13
Sum of the aliquot divisors of 9 = 4
and 13 – 4 = 9 = 3^2

Sum of the aliquot divisors of 34 = 20
Sum of the aliquot divisors of 9 = 4
and 20 – 4 = 16 = 4^2

Sum of the aliquot divisors of 35 = 13
Sum of the aliquot divisors of 9 = 4
and 13 – 4 = 9 = 3^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 10 = 8
and 17 – 8 = 9 = 3^2

Sum of the aliquot divisors of 14 = 10
Sum of the aliquot divisors of 11 = 1
and 10 – 1 = 9 = 3^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 11 = 1
and 17 – 1 = 16 = 4^2

Sum of the aliquot divisors of 40 = 50
Sum of the aliquot divisors of 11 = 1
and 50 – 1 = 49 = 7^2

Sum of the aliquot divisors of 34 = 20
Sum of the aliquot divisors of 12 = 16
and 20 – 16 = 4 = 2^2

Sum of the aliquot divisors of 14 = 10
Sum of the aliquot divisors of 13 = 1
and 10 – 1 = 9 = 3^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 13 = 1
and 17 – 1 = 16 = 4^2

Sum of the aliquot divisors of 40 = 50
Sum of the aliquot divisors of 13 = 1
and 50 – 1 = 49 = 7^2

Sum of the aliquot divisors of 22 = 14
Sum of the aliquot divisors of 14 = 10
and 14 – 10 = 4 = 2^2

Sum of the aliquot divisors of 27 = 13
Sum of the aliquot divisors of 15 = 9
and 13 – 9 = 4 = 2^2

Sum of the aliquot divisors of 35 = 13
Sum of the aliquot divisors of 15 = 9
and 13 – 9 = 4 = 2^2

Sum of the aliquot divisors of 32 = 31
Sum of the aliquot divisors of 16 = 15
and 31 – 15 = 16 = 4^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 17 = 1
and 17 – 1 = 16 = 4^2

Sum of the aliquot divisors of 40 = 50
Sum of the aliquot divisors of 17 = 1
and 50 – 1 = 49 = 7^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 19 = 1
and 17 – 1 = 16 = 4^2

Sum of the aliquot divisors of 40 = 50
Sum of the aliquot divisors of 19 = 1
and 50 – 1 = 49 = 7^2

Sum of the aliquot divisors of 32 = 31
Sum of the aliquot divisors of 20 = 22
and 31 – 22 = 9 = 3^2

Sum of the aliquot divisors of 24 = 36
Sum of the aliquot divisors of 21 = 11
and 36 – 11 = 25 = 5^2

Sum of the aliquot divisors of 33 = 15
Sum of the aliquot divisors of 21 = 11
and 15 – 11 = 4 = 2^2

Sum of the aliquot divisors of 34 = 20
Sum of the aliquot divisors of 21 = 11
and 20 – 11 = 9 = 3^2

Sum of the aliquot divisors of 40 = 50
Sum of the aliquot divisors of 22 = 14
and 50 – 14 = 36 = 6^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 23 = 1
and 17 – 1 = 16 = 4^2

Sum of the aliquot divisors of 40 = 50
Sum of the aliquot divisors of 23 = 1
and 50 – 1 = 49 = 7^2

Sum of the aliquot divisors of 30 = 42
Sum of the aliquot divisors of 25 = 6
and 42 – 6 = 36 = 6^2

Sum of the aliquot divisors of 32 = 31
Sum of the aliquot divisors of 25 = 6
and 31 – 6 = 25 = 5^2

Sum of the aliquot divisors of 33 = 15
Sum of the aliquot divisors of 25 = 6
and 15 – 6 = 9 = 3^2

Sum of the aliquot divisors of 36 = 55
Sum of the aliquot divisors of 25 = 6
and 55 – 6 = 49 = 7^2

Sum of the aliquot divisors of 38 = 22
Sum of the aliquot divisors of 25 = 6
and 22 – 6 = 16 = 4^2

Sum of the aliquot divisors of 34 = 20
Sum of the aliquot divisors of 26 = 16
and 20 – 16 = 4 = 2^2

Sum of the aliquot divisors of 38 = 22
Sum of the aliquot divisors of 27 = 13
and 22 – 13 = 9 = 3^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 27 = 13
and 17 – 13 = 4 = 2^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 29 = 1
and 17 – 1 = 16 = 4^2

Sum of the aliquot divisors of 40 = 50
Sum of the aliquot divisors of 29 = 1
and 50 – 1 = 49 = 7^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 31 = 1
and 17 – 1 = 16 = 4^2

Sum of the aliquot divisors of 40 = 50
Sum of the aliquot divisors of 31 = 1
and 50 – 1 = 49 = 7^2

Sum of the aliquot divisors of 38 = 22
Sum of the aliquot divisors of 35 = 13
and 22 – 13 = 9 = 3^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 35 = 13
and 17 – 13 = 4 = 2^2

Sum of the aliquot divisors of 39 = 17
Sum of the aliquot divisors of 37 = 1
and 17 – 1 = 16 = 4^2

Sum of the aliquot divisors of 40 = 50
Sum of the aliquot divisors of 37 = 1
and 50 – 1 = 49 = 7^2

Paul.

2. osedax says:

359 solutions for [1-100]. Largest pair, 12^2: [90, 1]
14,171 solutions for [1-1,000]. Largest pairs (5), 45^2: [840, 16], [840, 33], [960, 64], [960, 177], [960, 817]
449,512 solutions for [1-10,000]. Largest pair, 159^2: [9240, 217]

143 solutions for [1-100]. Largest pairs (45), 3^3: [24, 15] up to [100, 78]
1,557 solutions for [1-1,000]. Largest pairs (395), 3^3: [24, 15] up to [999, 982]
17,786 solutions for [1-10,000]. Largest pairs (4,310), 3^3: [24, 15] up to [9995, 3422]

The question is, what is the first pair for X^3 where X > 3?

Solutions for [1-100] for X^2 below.
[[2, 1], [3, 1], [5, 1], [7, 1], [8, 4], [8, 6], [9, 1], [9, 4], [10, 8], [10, 9], [11, 1], [12, 1], [12, 8], [13, 1], [14, 2], [14, 3], [14, 5], [14, 6], [14, 7], [14, 11], [14, 13], [15, 1], [15, 10], [16, 6], [17, 1], [19, 1], [20, 6], [20, 18], [21, 8], [21, 14], [22, 14], [23, 1], [24, 1], [24, 21], [26, 1], [26, 8], [26, 16], [27, 9], [27, 15], [28, 4], [29, 1], [30, 6], [30, 25], [31, 1], [32, 6], [32, 16], [32, 20], [32, 25], [33, 6], [33, 21], [33, 22], [33, 25], [34, 9], [34, 12], [34, 21], [34, 26], [35, 9], [35, 15], [36, 6], [36, 25], [37, 1], [38, 6], [38, 18], [38, 25], [38, 27], [38, 35], [39, 2], [39, 3], [39, 5], [39, 7], [39, 10], [39, 11], [39, 12], [39, 13], [39, 17], [39, 19], [39, 23], [39, 26], [39, 27], [39, 29], [39, 31], [39, 35], [39, 37], [40, 2], [40, 3], [40, 5], [40, 7], [40, 11], [40, 13], [40, 17], [40, 19], [40, 22], [40, 23], [40, 29], [40, 31], [40, 37], [41, 1], [42, 40], [43, 1], [44, 9], [44, 16], [44, 24], [44, 32], [44, 33], [45, 10], [45, 39], [46, 2], [46, 3], [46, 5], [46, 7], [46, 11], [46, 13], [46, 14], [46, 17], [46, 19], [46, 20], [46, 23], [46, 29], [46, 31], [46, 37], [46, 38], [46, 39], [46, 41], [46, 43], [47, 1], [48, 44], [49, 8], [49, 9], [50, 8], [50, 30], [51, 34], [51, 39], [52, 14], [52, 18], [52, 30], [52, 51], [53, 1], [54, 39], [54, 40], [55, 2], [55, 3], [55, 5], [55, 7], [55, 10], [55, 11], [55, 12], [55, 13], [55, 17], [55, 19], [55, 23], [55, 26], [55, 27], [55, 29], [55, 31], [55, 35], [55, 37], [55, 41], [55, 43], [55, 47], [55, 49], [55, 53], [56, 1], [56, 16], [56, 28], [56, 33], [56, 36], [57, 8], [57, 20], [57, 22], [57, 38], [58, 8], [58, 12], [58, 26], [58, 28], [58, 32], [58, 57], [59, 1], [60, 10], [60, 49], [61, 1], [62, 15], [62, 45], [63, 12], [63, 26], [63, 44], [63, 58], [64, 22], [64, 42], [65, 4], [65, 14], [65, 16], [65, 33], [66, 22], [66, 30], [67, 1], [68, 15], [68, 20], [68, 30], [68, 38], [68, 42], [68, 45], [69, 21], [69, 46], [69, 57], [70, 14], [70, 68], [71, 1], [72, 30], [72, 57], [72, 70], [73, 1], [74, 9], [74, 16], [74, 24], [74, 32], [74, 33], [75, 1], [75, 27], [75, 35], [75, 44], [75, 45], [75, 74], [76, 1], [76, 16], [76, 28], [76, 33], [76, 36], [76, 64], [77, 4], [77, 14], [77, 16], [77, 33], [78, 15], [78, 42], [78, 46], [78, 63], [78, 70], [79, 1], [80, 6], [80, 25], [80, 30], [80, 78], [81, 9], [81, 16], [81, 24], [81, 32], [81, 33], [82, 10], [82, 28], [82, 44], [82, 49], [82, 50], [82, 65], [82, 74], [82, 77], [82, 81], [83, 1], [84, 44], [84, 48], [84, 65], [84, 74], [84, 77], [84, 81], [85, 8], [85, 20], [85, 22], [85, 38], [85, 65], [85, 77], [86, 14], [86, 18], [86, 30], [86, 51], [87, 10], [87, 39], [87, 49], [87, 55], [87, 58], [88, 21], [88, 28], [88, 48], [88, 50], [89, 1], [90, 1], [90, 57], [90, 60], [90, 64], [90, 82], [90, 84], [90, 85], [91, 34], [91, 39], [91, 55], [92, 44], [92, 69], [92, 74], [92, 81], [93, 14], [93, 32], [93, 46], [93, 62], [93, 65], [93, 77], [94, 2], [94, 3], [94, 5], [94, 7], [94, 11], [94, 13], [94, 17], [94, 19], [94, 22], [94, 23], [94, 29], [94, 31], [94, 37], [94, 41], [94, 43], [94, 47], [94, 52], [94, 53], [94, 59], [94, 61], [94, 62], [94, 63], [94, 67], [94, 71], [94, 73], [94, 75], [94, 79], [94, 83], [94, 86], [94, 89], [95, 1], [95, 12], [95, 15], [95, 18], [95, 26], [95, 51], [95, 91], [96, 84], [96, 88], [96, 93], [97, 1], [98, 15], [98, 56], [98, 76], [99, 10], [99, 18], [99, 49], [99, 51], [99, 58], [99, 63], [99, 91], [100, 24], [100, 39], [100, 55], [100, 60], [100, 88]]

• benvitalis says:

Welcome to my site

• osedax says:

Thank you. I’ve solved many of these wonderful problems in the past, but have never commented before. I use them as little brain exercises for Python.