Table of pairs whose sum is one of the given square numbers

2023-04-07

The following table lists all 93 pairs which can be formed with the numbers from 1 to 60 whose sum is an element of the set {9, 36, 49, 64, 81}:

The first column lists the numbers i from 1 to 60.
The second column lists all the numbers greater than i with which i can be paired. That’s the column used by the first (slow) version of the python program.
The third column lists all the numbers smaller than i with which i can be paired. That’s the column used by the second (backward) version of the python program.
The fourth column lists all the numbers with which i can be paired, smaller or greater. That’s the column used by the third (fast forward) version of the python program.

i	higher pairs	lower pairs	all pairs
1	8, 35, 48		8, 35, 48
2	7, 34, 47		7, 34, 47
3	6, 33, 46		6, 33, 46
4	5, 32, 45, 60		5, 32, 45, 60
5	31, 44, 59	4	4, 31, 44, 59
6	30, 43, 58	3	3, 30, 43, 58
7	29, 42, 57	2	2, 29, 42, 57
8	28, 41, 56	1	1, 28, 41, 56
9	27, 40, 55		27, 40, 55
10	26, 39, 54		26, 39, 54
11	25, 38, 53		25, 38, 53
12	24, 37, 52		24, 37, 52
13	23, 36, 51		23, 36, 51
14	22, 35, 50		22, 35, 50
15	21, 34, 49		21, 34, 49
16	20, 33, 48		20, 33, 48
17	19, 32, 47		19, 32, 47
18	31, 46		31, 46
19	30, 45	17	17, 30, 45
20	29, 44	16	16, 29, 44
21	28, 43, 60	15	15, 28, 43, 60
22	27, 42, 59	14	14, 27, 42, 59
23	26, 41, 58	13	13, 26, 41, 58
24	25, 40, 57	12	12, 25, 40, 57
25	39, 56	24, 11	11, 24, 39, 56
26	38, 55	23, 10	10, 23, 38, 55
27	37, 54	22, 9	9, 22, 37, 54
28	36, 53	21, 8	8, 21, 36, 53
29	35, 52	20, 7	7, 20, 35, 52
30	34, 51	19, 6	6, 19, 34, 51
31	33, 50	18, 5	5, 18, 33, 50
32	49	17, 4	4, 17, 49
33	48	31, 16, 3	3, 16, 31, 48
34	47	30, 15, 2	2, 15, 30, 47
35	46	29, 14, 1	1, 14, 29, 46
36	45	28, 13	13, 28, 45
37	44	27, 12	12, 27, 44
38	43	26, 11	11, 26, 43
39	42	25, 10	10, 25, 42
40	41	24, 9	9, 24, 41
41		40, 23, 8	8, 23, 40
42		39, 22, 7	7, 22, 39
43		38, 21, 6	6, 21, 38
44		37, 20, 5	5, 20, 37
45		36, 19, 4	4, 19, 36
46		35, 18, 3	3, 18, 35
47		34, 17, 2	2, 17, 34
48		33, 16, 1	1, 16, 33
49		32, 15	15, 32
50		31, 14	14, 31
51		30, 13	13, 30
52		29, 12	12, 29
53		28, 11	11, 28
54		27, 10	10, 27
55		26, 9	9, 26
56		25, 8	8, 25
57		24, 7	7, 24
58		23, 6	6, 23
59		22, 5	5, 22
60		21, 4	4, 21