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Heinz numbers of integer partitions where every partial run (consecutive constant subsequence) has a different sum, and these sums include every integer from 0 to the greatest part.
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%I #13 Jun 11 2022 07:54:50

%S 1,2,4,6,8,16,20,30,32,56,64,90,128,140,176,210,256,416,512,616,990,

%T 1024,1088,1540,2048,2288,2310,2432,2970,4096,4950,5888,7072,7700,

%U 8008,8192,11550,12870,14848,16384,20020,20672,30030,31744,32768,38896,50490,55936

%N Heinz numbers of integer partitions where every partial run (consecutive constant subsequence) has a different sum, and these sums include every integer from 0 to the greatest part.

%C 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 Related concepts:

%C - A partition whose submultiset sums cover an initial interval is said to be complete (A126796, ranked by A325781).

%C - In a knapsack partition (A108917, ranked by A299702), every submultiset has a different sum.

%C - A complete partition that is also knapsack is said to be perfect (A002033, ranked by A325780).

%C - A partition whose partial runs have all different sums is said to be rucksack (A353864, ranked by A353866, complement A354583).

%e The terms together with their prime indices begin:

%e 1: {}

%e 2: {1}

%e 4: {1,1}

%e 6: {1,2}

%e 8: {1,1,1}

%e 16: {1,1,1,1}

%e 20: {1,1,3}

%e 30: {1,2,3}

%e 32: {1,1,1,1,1}

%e 56: {1,1,1,4}

%e 64: {1,1,1,1,1,1}

%e 90: {1,2,2,3}

%e 128: {1,1,1,1,1,1,1}

%e 140: {1,1,3,4}

%e 176: {1,1,1,1,5}

%e 210: {1,2,3,4}

%e 256: {1,1,1,1,1,1,1,1}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t norqQ[m_]:=Sort[m]==Range[0,Max[m]];

%t msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];

%t Select[Range[1000],norqQ[Total/@Select[msubs[primeMS[#]],SameQ@@#&]]&]

%Y Knapsack partitions are counted by A108917, ranked by A299702.

%Y Complete partitions are counted by A126796, ranked by A325781.

%Y These partitions are counted by A353865.

%Y This is a special case of A353866, counted by A353864, complement A354583.

%Y A001222 counts prime factors, distinct A001221.

%Y A056239 adds up prime indices, row sums of A112798 and A296150.

%Y A073093 counts prime-power divisors.

%Y A124010 gives prime signature, sorted A118914.

%Y A300273 ranks collapsible partitions, counted by A275870.

%Y A353832 represents the operation of taking run-sums of a partition.

%Y A353833 ranks partitions with all equal run-sums, nonprime A353834.

%Y A353836 counts partitions by number of distinct run-sums.

%Y A353852 ranks compositions with all distinct run-sums, counted by A353850.

%Y A353863 counts partitions whose weak run-sums cover an initial interval.

%Y Cf. A018818, A181819, A182857, A304442, A316413, A325862, A353835, A353838, A353839, A353861, A353931.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 07 2022