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Number of cross-balanced factorizations of n.
23

%I #16 Jun 19 2024 16:17:50

%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 1,1,1,6,1,1,2,2,1,1,1,4,1,1,1,3,1,1,1,3,1,3,1,2,1,1,1,5,1,2,2,5,1,1,1,3,1

%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).

%H Antti Karttunen, <a href="/A340654/b340654.txt">Table of n, a(n) for n = 1..65537</a>

%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}]

%o (PARI) A340654(n, m=n, om=omega(n),mbo=0) = if(1==n,(mbo==om), sumdiv(n, d, if((d>1)&&(d<=m), A340654(n/d, d, om, max(mbo,bigomega(d)))))); \\ _Antti Karttunen_, Jun 19 2024

%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

%E Data section extended up to a(105) by _Antti Karttunen_, Jun 19 2024