A385213 Number of maximal runs of consecutive parts increasing by 1 in the prime indices of n (with multiplicity).

0, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 3, 2, 2, 1, 3, 2, 2, 3, 3, 1, 1, 1, 5, 2, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 3, 2, 2, 1, 4, 2, 3, 2, 3, 1, 3, 2, 4, 2, 2, 1, 2, 1, 2, 3, 6, 2, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 2, 1, 5, 4, 2, 1, 3, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Jun 22 2025

Keywords

• nonn

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.

Examples

			The prime indices of 24 are {1,1,1,2}, with maximal runs ((1),(1),(1,2)), so a(24) = 3.
		

Crossrefs

Positions of first appearances are A000079.

For binary instead of prime indices we have A069010 (for anti-runs A384890).

For anti-runs instead of runs we have A384906.

A034839 counts subsets by number of maximal runs, for strict partitions A116674.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.

Cf. A002110, A048767, A130091, A300820, A356606, A351202, A382525, A384893.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Split[prix[n],#2==#1+1&]],{n,100}]