A353866 - OEIS

%I #7 Jun 10 2022 07:36:15

%S 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 29,30,31,32,33,34,35,37,38,39,41,42,43,44,45,46,47,49,50,51,52,53,54,

%U 55,56,57,58,59,61,62,64,65,66,67,68,69,70,71,73,74,75

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

%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 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 The terms together with their prime indices begin:

%e     1: {}

%e     2: {1}

%e     3: {2}

%e     4: {1,1}

%e     5: {3}

%e     6: {1,2}

%e     7: {4}

%e     8: {1,1,1}

%e     9: {2,2}

%e    10: {1,3}

%e    11: {5}

%e    13: {6}

%e    14: {1,4}

%e    15: {2,3}

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

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

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

%t Select[Range[100],UnsameQ@@Total/@Select[msubs[primeMS[#]],SameQ@@#&]&]

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

%Y The strong case is A353838, counted by A353837, complement A353839.

%Y These partitions are counted by A353864.

%Y The complete case is A353867, counted by A353865.

%Y The complement is A354583.

%Y A000041 counts partitions, strict A000009.

%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 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, A067340, A181819, A304442, A316413, A325862, A353833, A353835, A353861, A353931.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 06 2022