%I #12 Jan 19 2021 21:52:28

%S 1,1,1,0,1,0,1,0,0,0,1,2,1,0,0,0,1,2,1,2,0,0,1,1,0,0,0,2,1,0,1,0,0,0,

%T 0,2,1,0,0,1,1,0,1,2,2,0,1,0,0,2,0,2,1,1,0,1,0,0,1,0,1,0,2,0,0,0,1,2,

%U 0,0,1,0,1,0,2,2,0,0,1,0,0,0,1,0,0,0,0

%N Number of twice-balanced factorizations of n.

%C We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:

%C (1) the number of factors;

%C (2) the maximum image of A001222 over the factors;

%C (3) A001221(n).

%e The twice-balanced factorizations for n = 12, 120, 360, 480, 900, 2520:

%e 2*6 3*5*8 5*8*9 2*8*30 2*6*75 2*2*7*90

%e 3*4 2*2*30 2*4*45 3*8*20 2*9*50 2*3*5*84

%e 2*3*20 2*6*30 4*4*30 3*4*75 2*3*7*60

%e 2*5*12 2*9*20 4*6*20 3*6*50 2*5*7*36

%e 3*4*30 4*8*15 4*5*45 3*3*5*56

%e 3*6*20 5*8*12 5*6*30 3*3*7*40

%e 3*8*15 6*8*10 5*9*20 3*5*7*24

%e 4*5*18 2*12*20 2*10*45 2*2*2*315

%e 5*6*12 4*10*12 2*15*30 2*2*3*210

%e 2*10*18 2*18*25 2*2*5*126

%e 2*12*15 3*10*30 2*3*3*140

%e 3*10*12 3*12*25

%e 3*15*20

%e 5*10*18

%e 5*12*15

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Select[facs[n],#=={}||Length[#]==PrimeNu[n]==Max[PrimeOmega/@#]&]],{n,30}]

%Y The co-balanced version is A340596.

%Y The version for unlabeled multiset partitions is A340652.

%Y The balanced version is A340653.

%Y The cross-balanced version is A340654.

%Y Positions of zeros are A340656.

%Y Positions of nonzero terms are A340657.

%Y A001055 counts factorizations.

%Y A001221 counts distinct prime factors.

%Y A001222 counts prime factors with multiplicity.

%Y A045778 counts strict factorizations.

%Y A303975 counts distinct prime factors in prime indices.

%Y A316439 counts factorizations by product and length.

%Y Other balance-related sequences:

%Y - A010054 counts balanced strict partitions.

%Y - A047993 counts balanced partitions.

%Y - A098124 counts balanced compositions.

%Y - A106529 lists Heinz numbers of balanced partitions.

%Y - A340597 have an alt-balanced factorization.

%Y - A340598 counts balanced set partitions.

%Y - A340599 counts alt-balanced factorizations.

%Y - A340600 counts unlabeled balanced multiset partitions.

%Y Cf. A003963, A050336, A117409, A320655, A320656, A324518, A339846, A339890, A340607, A340608.

%K nonn

%O 1,12

%A _Gus Wiseman_, Jan 15 2021