A353842 - OEIS

%I #9 May 27 2022 16:48:22

%S 1,2,3,3,5,6,7,5,7,10,11,7,13,14,15,7,17,14,19,15,21,22,23,15,13,26,

%T 13,21,29,30,31,11,33,34,35,21,37,38,39,13,41,42,43,33,35,46,47,21,19,

%U 26,51,39,53,26,55,35,57,58,59,35,61,62,19,13,65,66,67,51

%N Last part of the trajectory of the partition run-sum transformation of n, using Heinz numbers.

%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 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 corresponds to the partitions (2,1,1) -> (2,2) -> (4).

%e The partition run-sum trajectory of 87780 is: 87780 -> 65835 -> 51205 -> 19855 -> 2915, so a(87780) = 2915.

%t Table[NestWhile[Times@@Prime/@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]&,n,!SquareFreeQ[#]&],{n,100}]

%Y The fixed points and image are A005117.

%Y For run-lengths instead of sums we have A304464/A304465, counted by A325268.

%Y These are the row-ends of A353840.

%Y Other sequences pertaining to partition trajectory are A353841-A353846.

%Y The version for compositions is A353855, run-ends of A353853.

%Y A001222 counts prime factors, distinct A001221.

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

%Y A182850 and A323014 give frequency depth.

%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, counted by A304442.

%Y A353835 counts distinct run-sums of prime indices, weak A353861.

%Y A353866 ranks rucksack partitions, counted by A353864.

%Y Cf. A071625, A073093, A181819, A325239, A325277, A353834, A353838, A353847, A353865, A353867.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 25 2022