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Irregular triangle read by rows where row k lists the run-sums of the multiset (weakly increasing sequence) of prime indices of n.
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%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