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%I #11 Jan 19 2021 21:52:20
%S 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,3,1,1,1,2,1,1,1,1,1,1,
%T 1,5,1,1,1,3,1,1,1,2,2,1,1,4,1,2,1,2,1,3,1,3,1,1,1,3,1,1,2,1,1,1,1,2,
%U 1,1,1,6,1,1,2,2,1,1,1,4,1,1,1,3,1,1,1
%N Number of cross-balanced factorizations of n.
%C We define a factorization of n into factors > 1 to be cross-balanced if either (1) it is empty or (2) the maximum image of A001222 over the factors is A001221(n).
%e The cross-balanced factorizations for n = 12, 24, 36, 72, 144, 240:
%e 2*6 4*6 4*9 2*4*9 4*4*9 8*30
%e 3*4 2*2*6 6*6 2*6*6 4*6*6 12*20
%e 2*3*4 2*2*9 3*4*6 2*2*4*9 5*6*8
%e 2*3*6 2*2*2*9 2*2*6*6 2*4*30
%e 3*3*4 2*2*3*6 2*3*4*6 2*6*20
%e 2*3*3*4 3*3*4*4 2*8*15
%e 2*2*2*2*9 3*4*20
%e 2*2*2*3*6 3*8*10
%e 2*2*3*3*4 4*5*12
%e 2*10*12
%e 2*3*5*8
%e 2*2*2*30
%e 2*2*3*20
%e 2*2*5*12
%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],#=={}||PrimeNu[n]==Max[PrimeOmega/@#]&]],{n,100}]
%Y Positions of terms > 1 are A126706.
%Y Positions of 1's are A303554.
%Y The co-balanced version is A340596.
%Y The version for unlabeled multiset partitions is A340651.
%Y The balanced version is A340653.
%Y The twice-balanced version is A340655.
%Y A001055 counts factorizations.
%Y A045778 counts strict factorizations.
%Y A316439 counts factorizations by product and length.
%Y A320655 counts factorizations into semiprimes.
%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 - A340652 counts unlabeled twice-balanced multiset partitions.
%Y - A340656 have no twice-balanced factorizations.
%Y - A340657 have a twice-balanced factorization.
%Y Cf. A003963, A117409, A303975, A320656, A324518, A339846, A339890, A340608.
%K nonn
%O 1,12
%A _Gus Wiseman_, Jan 15 2021
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