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A353866 Heinz numbers of rucksack partitions. Every prime-power divisor has a different sum of prime indices.

%I A353866 #7 Jun 10 2022 07:36:15
%S A353866 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,20,21,22,23,25,26,27,28,
%T A353866 29,30,31,32,33,34,35,37,38,39,41,42,43,44,45,46,47,49,50,51,52,53,54,
%U A353866 55,56,57,58,59,61,62,64,65,66,67,68,69,70,71,73,74,75
%N A353866 Heinz numbers of rucksack partitions. Every prime-power divisor has a different sum of prime indices.
%C A353866 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 A353866 In a knapsack partition (A108917, ranked by A299702), every submultiset has a different sum, so these are run-knapsack partitions or rucksack partitions for short.
%e A353866 The terms together with their prime indices begin:
%e A353866     1: {}
%e A353866     2: {1}
%e A353866     3: {2}
%e A353866     4: {1,1}
%e A353866     5: {3}
%e A353866     6: {1,2}
%e A353866     7: {4}
%e A353866     8: {1,1,1}
%e A353866     9: {2,2}
%e A353866    10: {1,3}
%e A353866    11: {5}
%e A353866    13: {6}
%e A353866    14: {1,4}
%e A353866    15: {2,3}
%e A353866    16: {1,1,1,1}
%e A353866 The sequence contains 18 because its prime-power divisors {1,2,3,9} have prime indices {}, {1}, {2}, {2,2} with distinct sums {0,1,2,4}. On the other hand, 12 is not in the sequence because {2} and {1,1} have the same sum.
%t A353866 msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
%t A353866 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A353866 Select[Range[100],UnsameQ@@Total/@Select[msubs[primeMS[#]],SameQ@@#&]&]
%Y A353866 Knapsack partitions are counted by A108917, ranked by A299702.
%Y A353866 The strong case is A353838, counted by A353837, complement A353839.
%Y A353866 These partitions are counted by A353864.
%Y A353866 The complete case is A353867, counted by A353865.
%Y A353866 The complement is A354583.
%Y A353866 A000041 counts partitions, strict A000009.
%Y A353866 A001222 counts prime factors, distinct A001221.
%Y A353866 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353866 A073093 counts prime-power divisors.
%Y A353866 A124010 gives prime signature, sorted A118914.
%Y A353866 A300273 ranks collapsible partitions, counted by A275870.
%Y A353866 A353832 represents the operation of taking run-sums of a partition.
%Y A353866 A353836 counts partitions by number of distinct run-sums.
%Y A353866 A353852 ranks compositions with all distinct run-sums, counted by A353850.
%Y A353866 A353863 counts partitions whose weak run-sums cover an initial interval.
%Y A353866 Cf. A018818, A067340, A181819, A304442, A316413, A325862, A353833, A353835, A353861, A353931.
%K A353866 nonn
%O A353866 1,2
%A A353866 _Gus Wiseman_, Jun 06 2022