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

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, 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, 5, 4, 3, 1, 9, 15, 4, 2, 8, 1, 6, 2, 7, 2, 6, 16

Offset

1,2

Comments

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.

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).

Links

Table of n, a(n) for n=1..82.

Example

Triangle begins:
  .
  1
  2
  2
  3
  1 2
  4
  3
  4
  1 3
  5
  2 2
  6
  1 4
  2 3
For example, the prime indices of 630 are {1,2,2,3,4}, so row 630 is (1,4,3,4).

Mathematica

Table[Cases[If[n==1, {}, FactorInteger[n]], {p_, k_}:>PrimePi[p]*k], {n, 30}]

Crossrefs

Positions of first appearances are A308495 plus 1.

The version for compositions is A353932, ranked by A353847.

Classes:

- singleton rows: A000961

- constant rows: A353833, nonprime A353834, counted by A304442

- strict rows: A353838, counted by A353837, complement A353839

Statistics:

- row lengths: A001221

- row sums: A056239

- row products: A304117

- row ranks (as partitions): A353832

- row image sizes: A353835

- row maxima: A353862

- row minima: A353931

A001222 counts prime factors with multiplicity.

A112798 and A296150 list partitions by rank.

A124010 gives prime signature, sorted A118914.

A300273 ranks collapsible partitions, counted by A275870.

A353840-A353846 pertain to partition run-sum trajectory.

A353861 counts distinct sums of partial runs of prime indices.

A353866 ranks rucksack partitions, counted by A353864.

Cf. A000040, A002110, A027748, A071625, A073093, A181819, A238279/A333755, A353850, A353852, A353867.

Sequence in context: A346264 A261789 A282972 * A088598 A200751 A077083

Adjacent sequences: A354581 A354582 A354583 * A354585 A354586 A354587

Keyword

nonn,tabf

Author

Gus Wiseman, Jun 17 2022

Status

approved