cp's OEIS Frontend

A374246 Number of prime factors of n counted with multiplicity (A001222) minus the greatest number of runs possible in a permutation of them (A373957).

%I A374246 #7 Jul 08 2024 16:41:41
%S A374246 0,0,0,1,0,0,0,2,1,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,1,0,2,0,0,0,0,4,0,0,
%T A374246 0,0,0,0,0,1,0,0,0,0,0,0,0,2,1,0,0,0,0,1,0,1,0,0,0,0,0,0,0,5,0,0,0,0,
%U A374246 0,0,0,0,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0
%N A374246 Number of prime factors of n counted with multiplicity (A001222) minus the greatest number of runs possible in a permutation of them (A373957).
%C A374246 a(n) = 0 iff n has separable prime factors (A335433). A multiset is separable iff it has a permutation that is an anti-run (meaning there are no adjacent equal parts).
%F A374246 a(n) = A001222(n) - A373957(n).
%e A374246 The runs of the 4 permutations of the prime factors of 24 are:
%e A374246   ((2,2,2),(3))
%e A374246   ((2,2),(3),(2))
%e A374246   ((2),(3),(2,2))
%e A374246   ((3),(2,2,2))
%e A374246 The longest have length 3, so a(24) = 4 - 3 = 1.
%t A374246 prifacs[n_]:=If[n==1,{}, Flatten[ConstantArray@@@FactorInteger[n]]];
%t A374246 Table[PrimeOmega[n]-Max@@Table[Length[Split[y]], {y,Permutations[prifacs[n]]}],{n,100}]
%Y A374246 Using the minimum instead of maximum number of runs gives A046660.
%Y A374246 Positions of first appearances are A151821 (powers of 2 except 2 itself).
%Y A374246 Positions of positive terms are A335448, complement A335433.
%Y A374246 This is an opposite version of A373957.
%Y A374246 The sister-sequence A374247 uses A001221 instead of A001222.
%Y A374246 This is the number of zeros at the end of row n of A374252.
%Y A374246 A001221 counts distinct prime factors, A001222 with multiplicity.
%Y A374246 A008480 counts permutations of prime factors.
%Y A374246 A027746 lists prime factors, row-sums A001414.
%Y A374246 A027748 is run-compression of prime factors, row-sums A008472.
%Y A374246 A304038 is run-compression of prime indices, row-sums A066328.
%Y A374246 A374250 maximizes sum of run-compression, for indices A373956.
%Y A374246 Cf. A003242, A007947, A026549, A056239, A124767, A246655, A316413, A333755.
%K A374246 nonn
%O A374246 1,8
%A A374246 _Gus Wiseman_, Jul 07 2024