cp's OEIS Frontend

A353867 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.

%I A353867 #13 Jun 11 2022 07:54:50
%S A353867 1,2,4,6,8,16,20,30,32,56,64,90,128,140,176,210,256,416,512,616,990,
%T A353867 1024,1088,1540,2048,2288,2310,2432,2970,4096,4950,5888,7072,7700,
%U A353867 8008,8192,11550,12870,14848,16384,20020,20672,30030,31744,32768,38896,50490,55936
%N A353867 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 A353867 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 A353867 Related concepts:
%C A353867 - A partition whose submultiset sums cover an initial interval is said to be complete (A126796, ranked by A325781).
%C A353867 - In a knapsack partition (A108917, ranked by A299702), every submultiset has a different sum.
%C A353867 - A complete partition that is also knapsack is said to be perfect (A002033, ranked by A325780).
%C A353867 - A partition whose partial runs have all different sums is said to be rucksack (A353864, ranked by A353866, complement A354583).
%e A353867 The terms together with their prime indices begin:
%e A353867     1: {}
%e A353867     2: {1}
%e A353867     4: {1,1}
%e A353867     6: {1,2}
%e A353867     8: {1,1,1}
%e A353867    16: {1,1,1,1}
%e A353867    20: {1,1,3}
%e A353867    30: {1,2,3}
%e A353867    32: {1,1,1,1,1}
%e A353867    56: {1,1,1,4}
%e A353867    64: {1,1,1,1,1,1}
%e A353867    90: {1,2,2,3}
%e A353867   128: {1,1,1,1,1,1,1}
%e A353867   140: {1,1,3,4}
%e A353867   176: {1,1,1,1,5}
%e A353867   210: {1,2,3,4}
%e A353867   256: {1,1,1,1,1,1,1,1}
%t A353867 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A353867 norqQ[m_]:=Sort[m]==Range[0,Max[m]];
%t A353867 msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
%t A353867 Select[Range[1000],norqQ[Total/@Select[msubs[primeMS[#]],SameQ@@#&]]&]
%Y A353867 Knapsack partitions are counted by A108917, ranked by A299702.
%Y A353867 Complete partitions are counted by A126796, ranked by A325781.
%Y A353867 These partitions are counted by A353865.
%Y A353867 This is a special case of A353866, counted by A353864, complement A354583.
%Y A353867 A001222 counts prime factors, distinct A001221.
%Y A353867 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353867 A073093 counts prime-power divisors.
%Y A353867 A124010 gives prime signature, sorted A118914.
%Y A353867 A300273 ranks collapsible partitions, counted by A275870.
%Y A353867 A353832 represents the operation of taking run-sums of a partition.
%Y A353867 A353833 ranks partitions with all equal run-sums, nonprime A353834.
%Y A353867 A353836 counts partitions by number of distinct run-sums.
%Y A353867 A353852 ranks compositions with all distinct run-sums, counted by A353850.
%Y A353867 A353863 counts partitions whose weak run-sums cover an initial interval.
%Y A353867 Cf. A018818, A181819, A182857, A304442, A316413, A325862, A353835, A353838, A353839, A353861, A353931.
%K A353867 nonn
%O A353867 1,2
%A A353867 _Gus Wiseman_, Jun 07 2022