%I #6 Jun 11 2022 07:55:00

%S 0,1,2,2,3,1,4,3,4,1,5,2,6,1,2,4,7,1,8,2,2,1,9,2,6,1,6,2,10,1,11,5,2,

%T 1,3,2,12,1,2,3,13,1,14,2,3,1,15,2,8,1,2,2,16,1,3,3,2,1,17,2,18,1,4,6,

%U 3,1,19,2,2,1,20,3,21,1,2,2,4,1,22,3,8,1

%N Least run-sum of the prime indices of n.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%C Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

%e The prime indices of 72 are {1,1,1,2,2}, with run-sums {3,4}, so a(72) = 3.

%t Table[Min@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k],{n,100}]

%Y Positions of first appearances are A008578.

%Y For run-lengths instead of run-sums we have A051904, greatest A051903.

%Y For run-sums and binary expansion we have A144790, greatest A038374.

%Y For run-lengths and binary expansion we have A175597, greatest A043276.

%Y Distinct run-sums are counted by A353835, weak A353861.

%Y The greatest run-sum is given by A353862.

%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 A124010 gives prime signature, sorted A118914.

%Y A304442 counts partitions with all equal run-sums, compositions A353851.

%Y A353832 represents the operation of taking run-sums of a partition.

%Y A353833 ranks partitions with all equal run sums, nonprime A353834.

%Y A353838 ranks partitions with all distinct run-sums, counted by A353837.

%Y A353840-A353846 pertain to partition run-sum trajectory.

%Y Cf. A067340, A073093, A181819, A300273, A316413, A353839, A353866.

%K nonn

%O 1,3

%A _Gus Wiseman_, Jun 07 2022