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EXAMPLE
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The non-subset-sums of y = {2,2,3} are {1,6}, with maximum 6, so y is counted under a(6).
The a(1) = 1 through a(6) = 15 multisets:
{2} {3} {4} {5} {6} {7}
{1,3} {1,4} {1,5} {1,6} {1,7}
{2,2} {2,3} {2,4} {2,5}
{1,1,4} {1,1,5} {3,3} {3,4}
{1,2,5} {1,1,6} {1,1,7}
{1,1,1,5} {1,2,6} {1,2,7}
{1,3,3} {1,3,4}
{2,2,2} {2,2,3}
{1,1,1,6} {1,1,1,7}
{1,1,2,6} {1,1,2,7}
{1,1,1,1,6} {1,1,3,7}
{1,2,2,7}
{1,1,1,1,7}
{1,1,1,2,7}
{1,1,1,1,1,7}
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MATHEMATICA
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prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n, 2*n], Max@@nmz[#]==n&]], {n, 5}]
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