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 A340597 Numbers with an alt-balanced factorization. 17
 4, 12, 18, 27, 32, 48, 64, 72, 80, 96, 108, 120, 128, 144, 160, 180, 192, 200, 240, 256, 270, 288, 300, 320, 360, 384, 400, 405, 432, 448, 450, 480, 500, 540, 576, 600, 640, 648, 672, 675, 720, 750, 768, 800, 864, 896, 900, 960, 972, 1000, 1008, 1024, 1080 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS We define a factorization into factors > 1 to be alt-balanced if its length is equal to its greatest factor. LINKS Table of n, a(n) for n=1..53. EXAMPLE The sequence of terms together with their prime signatures begins: 4: (2) 180: (2,2,1) 450: (1,2,2) 12: (2,1) 192: (6,1) 480: (5,1,1) 18: (1,2) 200: (3,2) 500: (2,3) 27: (3) 240: (4,1,1) 540: (2,3,1) 32: (5) 256: (8) 576: (6,2) 48: (4,1) 270: (1,3,1) 600: (3,1,2) 64: (6) 288: (5,2) 640: (7,1) 72: (3,2) 300: (2,1,2) 648: (3,4) 80: (4,1) 320: (6,1) 672: (5,1,1) 96: (5,1) 360: (3,2,1) 675: (3,2) 108: (2,3) 384: (7,1) 720: (4,2,1) 120: (3,1,1) 400: (4,2) 750: (1,1,3) 128: (7) 405: (4,1) 768: (8,1) 144: (4,2) 432: (4,3) 800: (5,2) 160: (5,1) 448: (6,1) 864: (5,3) 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. MATHEMATICA facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]]; Select[Range[100], Select[facs[#], Length[#]==Max[#]&]!={}&] CROSSREFS Numbers with a balanced factorization are A100959. These factorizations are counted by A340599. The twice-balanced version is A340657. A001055 counts factorizations. A045778 counts strict factorizations. A316439 counts factorizations by product and length. Other balance-related sequences: - A010054 counts balanced strict partitions. - A047993 counts balanced partitions. - A098124 counts balanced compositions. - A106529 lists Heinz numbers of balanced partitions. - A340596 counts co-balanced factorizations. - A340598 counts balanced set partitions. - A340600 counts unlabeled balanced multiset partitions. - A340653 counts balanced factorizations. - A340654 counts cross-balanced factorizations. - A340655 counts twice-balanced factorizations. Cf. A006141, A064174, A117409, A200750, A303975, A324518, A324522, A325134, A340607, A340608, A340611, A340656. Sequence in context: A301252 A057311 A063679 * A325234 A075867 A071929 Adjacent sequences: A340594 A340595 A340596 * A340598 A340599 A340600 KEYWORD nonn AUTHOR Gus Wiseman, Jan 15 2021 STATUS approved

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Last modified May 28 22:21 EDT 2023. Contains 363028 sequences. (Running on oeis4.)