cp's OEIS Frontend

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

%I A373957 #7 Jul 08 2024 16:41:50
%S A373957 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 A373957 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 A373957 2,3,1,5,1,2,3,3,2,3,1,3,1,2,1,4,2,2,2
%N A373957 Greatest number of runs in a permutation of the prime factors of n.
%C A373957 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 A373957 a(n) = A374247(n) - A001221(n).
%F A373957 a(n) = A001222(n) - A374246(n).
%e A373957 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 A373957   ((2,2,2),(3))
%e A373957   ((2,2),(3),(2))
%e A373957   ((2),(3),(2,2))
%e A373957   ((3),(2,2,2))
%e A373957 with lengths (2,3,3,2), with maximum a(24) = 3.
%t A373957 prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
%t A373957 Table[Max@@Table[Length[Split[y]],{y,Permutations[prifacs[n]]}],{n,100}]
%Y A373957 The minimum instead of maximum is A001221.
%Y A373957 Positions of 2 are A006881.
%Y A373957 Positions of first appearances appear to be A026549.
%Y A373957 Positions of 1 are A246655.
%Y A373957 The variation A374246 is the difference from bigomega (A001222).
%Y A373957 The variation A374247 is the difference with omega (A001221).
%Y A373957 This is the last position of a positive term in row n of A374252.
%Y A373957 A001221 counts distinct prime factors, A001222 with multiplicity.
%Y A373957 A008480 counts permutations of prime factors.
%Y A373957 A056239 adds up prime indices, row sums of A112798.
%Y A373957 A124767 counts runs in standard compositions, anti-runs A333381.
%Y A373957 A304038 is run-compression of prime indices, sums A066328, factors A027748.
%Y A373957 A333755 counts compositions by number of runs.
%Y A373957 A335433 lists numbers whose prime factors are separable, complement A335448.
%Y A373957 Cf. A003242, A007947, A238130, A316413, A373948, A373949, A374250, A374251.
%K A373957 nonn
%O A373957 1,6
%A A373957 _Gus Wiseman_, Jul 06 2024