cp's OEIS Frontend

A354584 Irregular triangle read by rows where row k lists the run-sums of the multiset (weakly increasing sequence) of prime indices of n.

%I A354584 #10 Jun 17 2022 22:12:49
%S A354584 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 A354584 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 A354584 5,4,3,1,9,15,4,2,8,1,6,2,7,2,6,16
%N A354584 Irregular triangle read by rows where row k lists the run-sums of the multiset (weakly increasing sequence) of prime indices of n.
%C A354584 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 A354584 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 A354584 Triangle begins:
%e A354584   .
%e A354584   1
%e A354584   2
%e A354584   2
%e A354584   3
%e A354584   1 2
%e A354584   4
%e A354584   3
%e A354584   4
%e A354584   1 3
%e A354584   5
%e A354584   2 2
%e A354584   6
%e A354584   1 4
%e A354584   2 3
%e A354584 For example, the prime indices of 630 are {1,2,2,3,4}, so row 630 is (1,4,3,4).
%t A354584 Table[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k],{n,30}]
%Y A354584 Positions of first appearances are A308495 plus 1.
%Y A354584 The version for compositions is A353932, ranked by A353847.
%Y A354584 Classes:
%Y A354584 - singleton rows: A000961
%Y A354584 - constant rows: A353833, nonprime A353834, counted by A304442
%Y A354584 - strict rows: A353838, counted by A353837, complement A353839
%Y A354584 Statistics:
%Y A354584 - row lengths: A001221
%Y A354584 - row sums: A056239
%Y A354584 - row products: A304117
%Y A354584 - row ranks (as partitions): A353832
%Y A354584 - row image sizes: A353835
%Y A354584 - row maxima: A353862
%Y A354584 - row minima: A353931
%Y A354584 A001222 counts prime factors with multiplicity.
%Y A354584 A112798 and A296150 list partitions by rank.
%Y A354584 A124010 gives prime signature, sorted A118914.
%Y A354584 A300273 ranks collapsible partitions, counted by A275870.
%Y A354584 A353840-A353846 pertain to partition run-sum trajectory.
%Y A354584 A353861 counts distinct sums of partial runs of prime indices.
%Y A354584 A353866 ranks rucksack partitions, counted by A353864.
%Y A354584 Cf. A000040, A002110, A027748, A071625, A073093, A181819, A238279/A333755, A353850, A353852, A353867.
%K A354584 nonn,tabf
%O A354584 1,2
%A A354584 _Gus Wiseman_, Jun 17 2022