A373957 - OEIS

%I #7 Jul 08 2024 16:41:50

%S 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,

%T 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,

%U 2,3,1,5,1,2,3,3,2,3,1,3,1,2,1,4,2,2,2

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

%C 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).

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

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

%e 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:

%e   ((2,2,2),(3))

%e   ((2,2),(3),(2))

%e   ((2),(3),(2,2))

%e   ((3),(2,2,2))

%e with lengths (2,3,3,2), with maximum a(24) = 3.

%t prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];

%t Table[Max@@Table[Length[Split[y]],{y,Permutations[prifacs[n]]}],{n,100}]

%Y The minimum instead of maximum is A001221.

%Y Positions of 2 are A006881.

%Y Positions of first appearances appear to be A026549.

%Y Positions of 1 are A246655.

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

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

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

%Y A001221 counts distinct prime factors, A001222 with multiplicity.

%Y A008480 counts permutations of prime factors.

%Y A056239 adds up prime indices, row sums of A112798.

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

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

%Y A333755 counts compositions by number of runs.

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

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

%K nonn

%O 1,6

%A _Gus Wiseman_, Jul 06 2024