OFFSET
1,2
COMMENTS
A non-subset-sum of a multiset of positive integers summing to n is an element of {1..n} that is not the sum of any submultiset. A multiset is incomplete if it has at least one non-subset-sum.
EXAMPLE
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}
MATHEMATICA
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}]
CROSSREFS
These multisets have ranks A365830.
Counts appearances of n in the rank statistic A365920.
Column sums of A365921.
The strict case is A366129.
A325799 counts non-subset-sums of prime indices.
A365543 counts partitions with a submultiset summing to k.
A365661 counts strict partitions w/ a subset summing to k.
A365918 counts non-subset-sums of partitions.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 30 2023
STATUS
approved