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A353840 Trajectory of the partition run-sum transformation of n, using Heinz numbers.

%I A353840 #11 May 27 2022 16:48:37
%S A353840 1,2,3,4,3,5,6,7,8,5,9,7,10,11,12,9,7,13,14,15,16,7,17,18,14,19,20,15,
%T A353840 21,22,23,24,15,25,13,26,27,13,28,21,29,30,31,32,11,33,34,35,36,21,37,
%U A353840 38,39,40,25,13,41,42,43,44,33,45,35,46,47,48,21,49,19
%N A353840 Trajectory of the partition run-sum transformation of n, using Heinz numbers.
%C A353840 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 A353840 The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 given in row 12 corresponds to the partitions (2,1,1) -> (2,2) -> (4).
%C A353840 This is the iteration of the transformation f described by Kimberling at A237685.
%e A353840 Triangle begins:
%e A353840    1
%e A353840    2
%e A353840    3
%e A353840    4  3
%e A353840    5
%e A353840    6
%e A353840    7
%e A353840    8  5
%e A353840    9  7
%e A353840   10
%e A353840   11
%e A353840   12  9  7
%e A353840 Row 87780 is the following trajectory (left column), with prime indices shown on the right:
%e A353840   87780: {1,1,2,3,4,5,8}
%e A353840   65835: {2,2,3,4,5,8}
%e A353840   51205: {3,4,4,5,8}
%e A353840   19855: {3,5,8,8}
%e A353840    2915: {3,5,16}
%t A353840 Table[NestWhileList[Times@@Prime/@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&,n,Not@*SquareFreeQ],{n,30}]
%Y A353840 The version for run-lengths instead of sums is A325239 or A325277.
%Y A353840 This is the iteration of A353832, with composition version A353847.
%Y A353840 Row-lengths are A353841, counted by A353846.
%Y A353840 Final terms are A353842.
%Y A353840 Counting rows by final omega gives A353843.
%Y A353840 Rows ending in a prime number are A353844, counted by A353845.
%Y A353840 These sequences for compositions are A353853-A353859.
%Y A353840 A001222 counts prime factors, distinct A001221.
%Y A353840 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353840 A124010 gives prime signature, sorted A118914.
%Y A353840 A182850 or A323014 gives frequency depth.
%Y A353840 A300273 ranks collapsible partitions, counted by A275870.
%Y A353840 A353833 ranks partitions with all equal run-sums, counted by A304442.
%Y A353840 A353835 counts distinct run-sums of prime indices, weak A353861.
%Y A353840 A353838 ranks partitions with all distinct run-sums, counted by A353837.
%Y A353840 A353862 gives greatest run-sum of prime indices, least A353931.
%Y A353840 Cf. A071625, A073093, A181819, A323023, A353743, A353834, A353864, A353865, A353866, A353867.
%K A353840 nonn
%O A353840 1,2
%A A353840 _Gus Wiseman_, May 25 2022