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A365830 Heinz numbers of incomplete integer partitions, meaning not every number from 0 to A056239(n) is the sum of some submultiset.

%I A365830 #9 Sep 28 2023 22:22:21
%S A365830 3,5,7,9,10,11,13,14,15,17,19,21,22,23,25,26,27,28,29,31,33,34,35,37,
%T A365830 38,39,41,43,44,45,46,47,49,50,51,52,53,55,57,58,59,61,62,63,65,66,67,
%U A365830 68,69,70,71,73,74,75,76,77,78,79,81,82,83,85,86,87,88,89
%N A365830 Heinz numbers of incomplete integer partitions, meaning not every number from 0 to A056239(n) is the sum of some submultiset.
%C A365830 First differs from A325798 in lacking 156.
%C A365830 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%C A365830 The complement (complete partitions) is A325781.
%e A365830 The terms together with their prime indices begin:
%e A365830    3: {2}
%e A365830    5: {3}
%e A365830    7: {4}
%e A365830    9: {2,2}
%e A365830   10: {1,3}
%e A365830   11: {5}
%e A365830   13: {6}
%e A365830   14: {1,4}
%e A365830   15: {2,3}
%e A365830   17: {7}
%e A365830   19: {8}
%e A365830   21: {2,4}
%e A365830   22: {1,5}
%e A365830   23: {9}
%e A365830   25: {3,3}
%e A365830   26: {1,6}
%e A365830   27: {2,2,2}
%e A365830   28: {1,1,4}
%e A365830 For example, the submultisets of (1,1,2,6) (right column) and their sums (left column) are:
%e A365830    0: ()
%e A365830    1: (1)
%e A365830    2: (2)  or (11)
%e A365830    3: (12)
%e A365830    4: (112)
%e A365830    6: (6)
%e A365830    7: (16)
%e A365830    8: (26) or (116)
%e A365830    9: (126)
%e A365830   10: (1126)
%e A365830 But 5 is missing, so 156 is in the sequence.
%t A365830 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A365830 nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
%t A365830 Select[Range[100],Length[nmz[prix[#]]]>0&]
%Y A365830 For prime indices instead of sums we have A080259, complement of A055932.
%Y A365830 The complement is A325781, counted by A126796, strict A188431.
%Y A365830 Positions of nonzero terms in A325799, complement A304793.
%Y A365830 These partitions are counted by A365924, strict A365831.
%Y A365830 A056239 adds up prime indices, row sums of A112798.
%Y A365830 A276024 counts positive subset-sums of partitions, strict A284640
%Y A365830 A299701 counts distinct subset-sums of prime indices.
%Y A365830 A365918 counts distinct non-subset-sums of partitions, strict A365922.
%Y A365830 A365923 counts partitions by distinct non-subset-sums, strict A365545.
%Y A365830 Cf. A001223, A046663, A073491, A098743, A304792, A365658, A365919.
%K A365830 nonn
%O A365830 1,1
%A A365830 _Gus Wiseman_, Sep 26 2023