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%I #9 Jan 19 2021 09:44:54
%S 4,12,18,27,32,48,64,72,80,96,108,120,128,144,160,180,192,200,240,256,
%T 270,288,300,320,360,384,400,405,432,448,450,480,500,540,576,600,640,
%U 648,672,675,720,750,768,800,864,896,900,960,972,1000,1008,1024,1080
%N Numbers with an alt-balanced factorization.
%C We define a factorization into factors > 1 to be alt-balanced if its length is equal to its greatest factor.
%e The sequence of terms together with their prime signatures begins:
%e 4: (2) 180: (2,2,1) 450: (1,2,2)
%e 12: (2,1) 192: (6,1) 480: (5,1,1)
%e 18: (1,2) 200: (3,2) 500: (2,3)
%e 27: (3) 240: (4,1,1) 540: (2,3,1)
%e 32: (5) 256: (8) 576: (6,2)
%e 48: (4,1) 270: (1,3,1) 600: (3,1,2)
%e 64: (6) 288: (5,2) 640: (7,1)
%e 72: (3,2) 300: (2,1,2) 648: (3,4)
%e 80: (4,1) 320: (6,1) 672: (5,1,1)
%e 96: (5,1) 360: (3,2,1) 675: (3,2)
%e 108: (2,3) 384: (7,1) 720: (4,2,1)
%e 120: (3,1,1) 400: (4,2) 750: (1,1,3)
%e 128: (7) 405: (4,1) 768: (8,1)
%e 144: (4,2) 432: (4,3) 800: (5,2)
%e 160: (5,1) 448: (6,1) 864: (5,3)
%e For example, there are two alt-balanced factorizations of 480, namely (2*3*4*4*5) and (2*2*2*2*5*6), so 480 in the sequence.
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t Select[Range[100],Select[facs[#],Length[#]==Max[#]&]!={}&]
%Y Numbers with a balanced factorization are A100959.
%Y These factorizations are counted by A340599.
%Y The twice-balanced version is A340657.
%Y A001055 counts factorizations.
%Y A045778 counts strict factorizations.
%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 - A340596 counts co-balanced factorizations.
%Y - A340598 counts balanced set partitions.
%Y - A340600 counts unlabeled balanced multiset partitions.
%Y - A340653 counts balanced factorizations.
%Y - A340654 counts cross-balanced factorizations.
%Y - A340655 counts twice-balanced factorizations.
%Y Cf. A006141, A064174, A117409, A200750, A303975, A324518, A324522, A325134, A340607, A340608, A340611, A340656.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jan 15 2021
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