A374247 The greatest number of runs possible in a permutation of the prime factors of n (A373957) minus the number of distinct such factors (A001221).

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

Offset: 1

Author

Gus Wiseman, Jul 07 2024

Keywords

• nonn

Comments

If n has separable prime factors (A335433), then a(n) = A001222(n) - A001221(n) = A046660(n). A multiset is separable iff it has an anti-run permutation (meaning there are no adjacent equal parts).

Examples

			The runs of the 6 permutations of the prime factors of 36 are:
  ((2,2),(3,3))
  ((2),(3),(2),(3))
  ((2),(3,3),(2))
  ((3),(2,2),(3))
  ((3),(2),(3),(2))
  ((3,3),(2,2))
The longest length is 4, so a(36) = 4 - 2 = 2.
		

Crossrefs

Positions of first appearances appear to be A026549.

Positions of nonzero terms are A126706, complement A303554.

This is an opposite version of A373957.

The sister-sequence A374246 uses A001222 instead of A001221.

This is the number of nonzero terms in row n of A374252.

A003242 counts run-compressed compositions, i.e., anti-runs.

A008480 counts permutations of prime factors, by number of runs A374252.

A027746 lists prime factors, row-sums A001414.

A027748 is run-compression of prime factors, row-sums A008472.

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

A333755 counts compositions by number of runs.

A335433 lists separable numbers, complement A335448.

A374250 maximizes sum of run-compression, for indices A373956.

Cf. A007947, A046660, A124767, A151821, A238130, A373951, A374251.

Programs

Mathematica

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

Formula

a(n) = A373957(n) - A001221(n).