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Sums of distinct primes

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Almost all integers can be represented as a sum of distinct primes, see A000586. This follows from Bertrand's postulate together with decompositions for n = 7 through 14. Checking smaller numbers shows that only 1, 4, and 6 lack such a representation.

THE SUMS OF 1s IN THE FOLLOWING CHART ARE ONLY PLACEHOLDERS. THE CHART IS NOT COMPLETE YET.

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