login
Numbers with an alt-balanced factorization.
17

%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