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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 |
