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%I #14 Oct 22 2023 15:13:38
%S 1,1,1,0,1,0,1,1,0,0,1,2,1,0,0,1,1,2,1,2,0,0,1,1,0,0,1,2,1,3,1,1,0,0,
%T 0,2,1,0,0,1,1,3,1,2,2,0,1,2,0,2,0,2,1,1,0,1,0,0,1,2,1,0,2,1,0,3,1,2,
%U 0,3,1,3,1,0,2,2,0,3,1,2,1,0,1,2,0,0,0,1,1,2,0,2,0,0,0,3,1,2,2,2,1,3,1,1,3,0,1,3,1,3,0,2,1,3,0,2,2,0,0,4
%N Number of balanced factorizations of n.
%C A factorization into factors > 1 is balanced if it is empty or its length is equal to its maximum Omega (A001222).
%H Antti Karttunen, <a href="/A340653/b340653.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%e The balanced factorizations for n = 120, 144, 192, 288, 432, 768:
%e 3*5*8 2*8*9 3*8*8 4*8*9 6*8*9 8*8*12
%e 2*2*30 3*6*8 4*6*8 6*6*8 2*8*27 2*2*8*24
%e 2*3*20 2*4*18 2*8*12 2*8*18 3*8*18 2*3*8*16
%e 2*5*12 2*6*12 4*4*12 3*8*12 4*4*27 2*4*4*24
%e 3*4*12 2*2*2*24 4*4*18 4*6*18 2*4*6*16
%e 2*2*3*16 4*6*12 4*9*12 3*4*4*16
%e 2*12*12 6*6*12 2*2*12*16
%e 2*2*2*36 2*12*18 2*2*2*2*48
%e 2*2*3*24 3*12*12 2*2*2*3*32
%e 2*3*3*16 2*2*2*54
%e 2*2*3*36
%e 2*3*3*24
%e 3*3*3*16
%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[#]==Max[PrimeOmega/@#]&]],{n,100}]
%o (PARI) A340653(n, m=n, mbo=0, e=0) = if(1==n, mbo==e, sumdiv(n, d, if((d>1)&&(d<=m), A340653(n/d, d, max(mbo,bigomega(d)), 1+e)))); \\ _Antti Karttunen_, Oct 22 2023
%Y Positions of zeros are A001358.
%Y Positions of nonzero terms are A100959.
%Y The co-balanced version is A340596.
%Y Taking maximum factor instead of maximum Omega gives A340599.
%Y The cross-balanced version is A340654.
%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 - A340600 counts unlabeled balanced multiset partitions.
%Y - A340656 have no twice-balanced factorizations.
%Y - A340657 have a twice-balanced factorization.
%Y Cf. A003963, A117409, A303975, A320656, A339846, A339890, A340608.
%K nonn
%O 1,12
%A _Gus Wiseman_, Jan 15 2021
%E Data section extended up to a(120) by _Antti Karttunen_, Oct 22 2023
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