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A340655 Number of twice-balanced factorizations of n.

%I A340655 #12 Jan 19 2021 21:52:28
%S A340655 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 A340655 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 A340655 0,0,1,0,1,0,2,2,0,0,1,0,0,0,1,0,0,0,0
%N A340655 Number of twice-balanced factorizations of n.
%C A340655 We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:
%C A340655 (1) the number of factors;
%C A340655 (2) the maximum image of A001222 over the factors;
%C A340655 (3) A001221(n).
%e A340655 The twice-balanced factorizations for n = 12, 120, 360, 480, 900, 2520:
%e A340655   2*6   3*5*8    5*8*9     2*8*30    2*6*75    2*2*7*90
%e A340655   3*4   2*2*30   2*4*45    3*8*20    2*9*50    2*3*5*84
%e A340655         2*3*20   2*6*30    4*4*30    3*4*75    2*3*7*60
%e A340655         2*5*12   2*9*20    4*6*20    3*6*50    2*5*7*36
%e A340655                  3*4*30    4*8*15    4*5*45    3*3*5*56
%e A340655                  3*6*20    5*8*12    5*6*30    3*3*7*40
%e A340655                  3*8*15    6*8*10    5*9*20    3*5*7*24
%e A340655                  4*5*18    2*12*20   2*10*45   2*2*2*315
%e A340655                  5*6*12    4*10*12   2*15*30   2*2*3*210
%e A340655                  2*10*18             2*18*25   2*2*5*126
%e A340655                  2*12*15             3*10*30   2*3*3*140
%e A340655                  3*10*12             3*12*25
%e A340655                                      3*15*20
%e A340655                                      5*10*18
%e A340655                                      5*12*15
%t A340655 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A340655 Table[Length[Select[facs[n],#=={}||Length[#]==PrimeNu[n]==Max[PrimeOmega/@#]&]],{n,30}]
%Y A340655 The co-balanced version is A340596.
%Y A340655 The version for unlabeled multiset partitions is A340652.
%Y A340655 The balanced version is A340653.
%Y A340655 The cross-balanced version is A340654.
%Y A340655 Positions of zeros are A340656.
%Y A340655 Positions of nonzero terms are A340657.
%Y A340655 A001055 counts factorizations.
%Y A340655 A001221 counts distinct prime factors.
%Y A340655 A001222 counts prime factors with multiplicity.
%Y A340655 A045778 counts strict factorizations.
%Y A340655 A303975 counts distinct prime factors in prime indices.
%Y A340655 A316439 counts factorizations by product and length.
%Y A340655 Other balance-related sequences:
%Y A340655 - A010054 counts balanced strict partitions.
%Y A340655 - A047993 counts balanced partitions.
%Y A340655 - A098124 counts balanced compositions.
%Y A340655 - A106529 lists Heinz numbers of balanced partitions.
%Y A340655 - A340597 have an alt-balanced factorization.
%Y A340655 - A340598 counts balanced set partitions.
%Y A340655 - A340599 counts alt-balanced factorizations.
%Y A340655 - A340600 counts unlabeled balanced multiset partitions.
%Y A340655 Cf. A003963, A050336, A117409, A320655, A320656, A324518, A339846, A339890, A340607, A340608.
%K A340655 nonn
%O A340655 1,12
%A A340655 _Gus Wiseman_, Jan 15 2021