%I #10 Jun 17 2022 22:12:49
%S 1,2,2,3,1,2,4,3,4,1,3,5,2,2,6,1,4,2,3,4,7,1,4,8,2,3,2,4,1,5,9,3,2,6,
%T 1,6,6,2,4,10,1,2,3,11,5,2,5,1,7,3,4,2,4,12,1,8,2,6,3,3,13,1,2,4,14,2,
%U 5,4,3,1,9,15,4,2,8,1,6,2,7,2,6,16
%N Irregular triangle read by rows where row k lists the run-sums of the multiset (weakly increasing sequence) of 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 Triangle begins:
%e .
%e 1
%e 2
%e 2
%e 3
%e 1 2
%e 4
%e 3
%e 4
%e 1 3
%e 5
%e 2 2
%e 6
%e 1 4
%e 2 3
%e For example, the prime indices of 630 are {1,2,2,3,4}, so row 630 is (1,4,3,4).
%t Table[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k],{n,30}]
%Y Positions of first appearances are A308495 plus 1.
%Y The version for compositions is A353932, ranked by A353847.
%Y Classes:
%Y - singleton rows: A000961
%Y - constant rows: A353833, nonprime A353834, counted by A304442
%Y - strict rows: A353838, counted by A353837, complement A353839
%Y Statistics:
%Y - row lengths: A001221
%Y - row sums: A056239
%Y - row products: A304117
%Y - row ranks (as partitions): A353832
%Y - row image sizes: A353835
%Y - row maxima: A353862
%Y - row minima: A353931
%Y A001222 counts prime factors with multiplicity.
%Y A112798 and A296150 list partitions by rank.
%Y A124010 gives prime signature, sorted A118914.
%Y A300273 ranks collapsible partitions, counted by A275870.
%Y A353840-A353846 pertain to partition run-sum trajectory.
%Y A353861 counts distinct sums of partial runs of prime indices.
%Y A353866 ranks rucksack partitions, counted by A353864.
%Y Cf. A000040, A002110, A027748, A071625, A073093, A181819, A238279/A333755, A353850, A353852, A353867.
%K nonn,tabf
%O 1,2
%A _Gus Wiseman_, Jun 17 2022