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A354583 Heinz numbers of non-rucksack partitions: not every prime-power divisor has a different sum of prime indices.

%I A354583 #10 Jun 17 2022 22:12:44
%S A354583 12,24,36,40,48,60,63,72,80,84,96,108,112,120,126,132,144,156,160,168,
%T A354583 180,189,192,200,204,216,224,228,240,252,264,276,280,288,300,312,315,
%U A354583 320,324,325,336,348,351,352,360,372,378,384,396,400,408,420,432,440
%N A354583 Heinz numbers of non-rucksack partitions: not every prime-power divisor has a different sum of prime indices.
%C A354583 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 A354583 The term rucksack is short for run-knapsack.
%e A354583 The terms together with their prime indices begin:
%e A354583    12: {1,1,2}
%e A354583    24: {1,1,1,2}
%e A354583    36: {1,1,2,2}
%e A354583    40: {1,1,1,3}
%e A354583    48: {1,1,1,1,2}
%e A354583    60: {1,1,2,3}
%e A354583    63: {2,2,4}
%e A354583    72: {1,1,1,2,2}
%e A354583    80: {1,1,1,1,3}
%e A354583    84: {1,1,2,4}
%e A354583    96: {1,1,1,1,1,2}
%e A354583   108: {1,1,2,2,2}
%e A354583   112: {1,1,1,1,4}
%e A354583   120: {1,1,1,2,3}
%e A354583   126: {1,2,2,4}
%e A354583   132: {1,1,2,5}
%e A354583   144: {1,1,1,1,2,2}
%e A354583   156: {1,1,2,6}
%e A354583   160: {1,1,1,1,1,3}
%e A354583   168: {1,1,1,2,4}
%e A354583 For example, {2,2,2,3,3} does not have distinct run-sums because 2+2+2 = 3+3, so 675 is in the sequence.
%t A354583 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A354583 Select[Range[100],!UnsameQ@@Total/@primeMS/@Select[Divisors[#],PrimePowerQ]&]
%Y A354583 Knapsack partitions are counted by A108917, ranked by A299702.
%Y A354583 Non-knapsack partitions are ranked by A299729.
%Y A354583 The non-partial version is A353839, complement A353838 (counted by A353837).
%Y A354583 The complement is A353866, counted by A353864.
%Y A354583 The complete complement is A353867, counted by A353865.
%Y A354583 The complement for compositions is counted by A354580.
%Y A354583 A001222 counts prime factors, distinct A001221.
%Y A354583 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A354583 A073093 counts prime-power divisors.
%Y A354583 A300273 ranks collapsible partitions, counted by A275870.
%Y A354583 A304442 counts partitions with all equal run-sums, ranked by A353833.
%Y A354583 A333223 ranks knapsack compositions, counted by A325676.
%Y A354583 A353852 ranks compositions with all distinct run-sums, counted by A353850.
%Y A354583 A353861 counts distinct partial run-sums of prime indices.
%Y A354583 A354584 lists run-sums of prime indices, rows ranked by A353832.
%Y A354583 Cf. A005811, A118914, A124010, A175413, A181819, A182857, A316413, A325862, A353834, A353835, A353836, A353931.
%K A354583 nonn
%O A354583 1,1
%A A354583 _Gus Wiseman_, Jun 15 2022