A373957 Greatest number of runs in a permutation of the prime factors of n.

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

Offset: 1

Author

Gus Wiseman, Jul 06 2024

Keywords

• nonn

Comments

If n belongs to A335433 (the separable case), then a(n) = A001222(n). A multiset is separable iff it has a permutation that is an anti-run (meaning there are no adjacent equal parts).

Examples

			The prime factors of 24 are {2,2,2,3}, with permutations (2,2,2,3), (2,2,3,2), (2,3,2,2), (3,2,2,2), with runs:
  ((2,2,2),(3))
  ((2,2),(3),(2))
  ((2),(3),(2,2))
  ((3),(2,2,2))
with lengths (2,3,3,2), with maximum a(24) = 3.
		

Crossrefs

The minimum instead of maximum is A001221.

Positions of 2 are A006881.

Positions of first appearances appear to be A026549.

Positions of 1 are A246655.

The variation A374246 is the difference from bigomega (A001222).

The variation A374247 is the difference with omega (A001221).

This is the last position of a positive term in row n of A374252.

A001221 counts distinct prime factors, A001222 with multiplicity.

A008480 counts permutations of prime factors.

A056239 adds up prime indices, row sums of A112798.

A124767 counts runs in standard compositions, anti-runs A333381.

A304038 is run-compression of prime indices, sums A066328, factors A027748.

A333755 counts compositions by number of runs.

A335433 lists numbers whose prime factors are separable, complement A335448.

Cf. A003242, A007947, A238130, A316413, A373948, A373949, A374250, A374251.

Programs

Mathematica

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Max@@Table[Length[Split[y]],{y,Permutations[prifacs[n]]}],{n,100}]

Formula

a(n) = A374247(n) - A001221(n).

a(n) = A001222(n) - A374246(n).