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%I #9 May 27 2022 16:48:28
%S 0,1,1,2,1,1,1,2,2,1,1,3,1,1,1,2,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,2,1,1,
%T 1,2,1,1,1,3,1,1,1,2,2,1,1,2,2,2,1,2,1,2,1,2,1,1,1,3,1,1,3,2,1,1,1,2,
%U 1,1,1,2,1,1,2,2,1,1,1,2,2,1,1,4,1,1,1
%N Length of the trajectory of the partition run-sum transformation of n, using Heinz numbers; a(1) = 0.
%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 Starting with n, this is one plus the number of times one must apply A353832 to reach a squarefree number.
%C Also Kimberling's depth statistic (defined in A237685 and A237750) plus one.
%e The trajectory for a(1080) = 4 is the following, with prime indices shown on the right:
%e 1080: {1,1,1,2,2,2,3}
%e 325: {3,3,6}
%e 169: {6,6}
%e 37: {12}
%e The trajectory for a(87780) = 5 is the following, with prime indices shown on the right:
%e 87780: {1,1,2,3,4,5,8}
%e 65835: {2,2,3,4,5,8}
%e 51205: {3,4,4,5,8}
%e 19855: {3,5,8,8}
%e 2915: {3,5,16}
%e The trajectory for a(39960) = 5 is the following, with prime indices shown on the right:
%e 39960: {1,1,1,2,2,2,3,12}
%e 12025: {3,3,6,12}
%e 6253: {6,6,12}
%e 1369: {12,12}
%e 89: {24}
%t Table[If[n==1,0,Length[NestWhileList[Times@@Prime/@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]&,n,!SquareFreeQ[#]&]]],{n,100}]
%Y Positions of 1's are A005117.
%Y The version for run-lengths instead of sums is A182850 or A323014.
%Y Positions of first appearances are A353743.
%Y These are the row-lengths of A353840.
%Y Other sequences pertaining to this trajectory are A353842-A353845.
%Y Counting partitions by this statistic gives A353846.
%Y The version for compositions is A353854, run-lengths of A353853.
%Y A001222 counts prime factors, distinct A001221.
%Y A005811 counts runs in binary expansion.
%Y A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A300273 ranks collapsible partitions, counted by A275870.
%Y A318928 gives runs-resistance of binary expansion.
%Y A353832 represents the operation of taking run-sums of a partition.
%Y A353833 ranks partitions with all equal run-sums, counted by A304442.
%Y A353835 counts distinct run-sums of prime indices, weak A353861.
%Y A353838 ranks partitions with all distinct run-sums, counted by A353837.
%Y A353866 ranks rucksack partitions, counted by A353864.
%Y Cf. A071625, A073093, A181819, A182857, A325239, A325277, A325278, A353834, A353847, A353865, A353867.
%K nonn
%O 1,4
%A _Gus Wiseman_, May 25 2022
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