A384906 Number of maximal anti-runs of consecutive parts not increasing by 1 in the prime indices of n (with multiplicity).

0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Jun 22 2025

Keywords

• nonn

Comments

First differs from A300820 at a(462) = 3, A300820(462) = 2.

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 462 are {1,2,4,5}, with maximal anti-runs ((1),(2,4),(5)), so a(462) = 3.
		

Crossrefs

For the strict case we have A356228.

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

For runs instead of anti-runs we have A385213.

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. A130091, A245562, A268193, A300820, A351202, A356607, A382525, A384177, A384321, 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}]