三重内积(增加难度)的最小值

王守恩

明确方向！来个估算公式。——Table[B*Floor[((n - 1)/2)^(n/B)], {B, 2, 6}, {n, B, 30}]——这个“1”有问题。

公式。{0, 2, 04, 10, 30, 092, 300, 1024, 3690, 13974, 55360, 228566, 980444, 4357778, 20022582, 94906264, 463233892, 2324522934, 11974738784, 63245553202, 342067871622, 1892543519452, 10700500210946},
B—2。{3, 5, 10, 22, 54, 142, 402, 1206, 3810, 12636, 43776, 157824, 590520, 2287080, 09148320, 37719360, 160029696, 0697553280, 03119552640, 14295585696, 067052240640, 0321571257120, 01575370944000——A127180

公式。{3, 3, 09, 18, 36, 084, 192, 450, 1095, 2745, 7062, 18651, 50421, 139350, 393216, 1131447, 3316323, 9891873, 30000000, 92436348, 289155753, 917706858, 2953237026, 9631073577, 31813498119, 106390559790},
B—3。{6, 9, 15, 28, 52, 103, 214, 462, 1026, 2348, 5520, 13304, 32808, 082668,

公式。{04, 08, 12, 24, 48, 088, 168, 332, 664, 1352, 2800, 5904, 12656, 27552, 60872, 136364, 309512, 711308, 1654248, 3891060, 9252240, 22230168, 53947964, 132183612, 326886036, 815620716, 2052674124},
B—4。{10, 14, 21, 35, 60, 100, 175, 319, 594, 1124, 2174, 4278,

公式。{10, 15, 20, 35, 60, 100, 170, 295, 525, 940, 1715, 3155, 5880, 11085, 21135, 40725, 79240, 155665, 308585, 617045, 1244160, 2528725, 5179215, 10686585, 22208160, 46470570},
B—5。{15, 20, 28, 43, 69, 110, 170, 273, 455, 772, 1334, 2308,——40#——B=1,2,3,4,5。

公式。{12, 18, 30, 48, 72, 114, 180, 288, 468, 774, 1290, 2172, 3684, 6306, 10890, 18972, 33300, 58902, 104940, 188250, 339960, 617868, 1129902, 2078592, 3845838}}
B—6。{},

1. Quiet@Needs["Combinatorica`"]; m = \[Infinity]; t = {}; Do[u = Total[Times @@@ v]; If[u < m, m = u; t = v];,
   
2. {v, Combinatorica`KSetPartitions[{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, 5]}]; {m, t, Times @@@ t}

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B—5。有这么一个算法。——我这里肯定是来不了“2308”。

王守恩

B—2。

(1)——A127180——有这么个公式。

1. Table[Module[{s, t}, {s, t} = MinimalBy[{#, Complement[Range[n], #]} & /@ Subsets[Range[n]], Abs[Times @@ #[[1]] - Times @@ #[[2]]] &][[1]]; Times @@ s + Times @@ t], {n, 3, 24}]

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{5, 10, 22, 54, 142, 402, 1206, 3810, 12636, 43776, 157824, 590520, 2287080, 9148320, 37719360, 160029696, 697553280, 3119552640, 14295585696, 67052240640, 321571257120, 1575370944000}

(2)——可以短,速度还快了。

1. Table[With[{t = n!}, Min[# + t/# &[Times @@ #] & /@ Subsets[Range[2, n], {1, Floor[n/2]}]]], {n, 3, 24}]

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{5, 10, 22, 54, 142, 402, 1206, 3810, 12636, 43776, 157824, 590520, 2287080, 9148320, 37719360, 160029696, 697553280, 3119552640, 14295585696, 67052240640, 321571257120, 1575370944000}

(3)——这个可以把明细列出来。

1. Table[Needs["Combinatorica`"]; m = Infinity; t = {}; Do[u = Total[Times @@@ v]; If[u < m, m = u; t = v];, {v, Combinatorica`KSetPartitions[Range[2, n], 2]}]; {m, t, Times @@@ t}, {n, 3, 24}]

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{5, {{2}, {3}}, {2, 3}},
{10, {{2, 3}, {4}}, {6, 4}},
{22, {{2, 5}, {3, 4}}, {10, 12}},
{54, {{2, 3, 4}, {5, 6}}, {24, 30}},
{142, {{2, 5, 7}, {3, 4, 6}}, {70, 72}},
{402, {{2, 3, 4, 8}, {5, 6, 7}}, {192, 210}},
{1206, {{2, 4, 8, 9}, {3, 5, 6, 7}}, {576, 630}},
{3810, {{2, 4, 5, 6, 8}, {3, 7, 9, 10}}, {1920, 1890}},
{12636, {{2, 5, 7, 9, 10}, {3, 4, 6, 8, 11}}, {6300, 6336}},
{43776, {{2, 3, 5, 8, 9, 10}, {4, 6, 7, 11, 12}}, {21600, 22176}},
{157824, {{2, 5, 8, 9, 10, 11}, {3, 4, 6, 7, 12, 13}}, {79200, 78624}},
{590520, {{2, 4, 6, 7, 8, 10, 11}, {3, 5, 9, 12, 13, 14}}, {295680, 294840}},
{2287080, {{2, 3, 4, 6, 8, 9, 10, 11}, {5, 7, 12, 13, 14 15}}, {1140480, 1146600}},
{9148320, {{2, 4, 6, 8, 9, 10, 11, 12}, {3, 5, 7, 13, 14, 15, 16}}, {4561920, 4586400}},
{37719360, {{2, 4, 5, 6, 7, 8, 9, 12, 13}, {3, 10, 11, 14, 15, 16, 17}}, {18869760, 18849600}},
{160029696, {{2, 3, 6, 8, 9, 11, 12, 13, 18}, {4, 5, 7, 10, 14, 15, 16, 17}}, {80061696, 79968000}},
{697553280, {{2, 5, 7, 8, 10, 13, 14, 18, 19}, {3, 4, 6, 9, 11, 12, 15, 16, 17}}, {348566400, 348986880}},
{3119552640, {{2, 3, 4, 8, 9, 11, 12, 18, 19, 20}, {5, 6, 7, 10, 13, 14, 15, 16, 17}}, {1560176640, 1559376000}},
{14295585696, {{2, 5, 7, 10, 11, 13, 14, 15, 17, 20}, {3, 4, 6, 8, 9, 12, 16, 18, 19, 21}}, {7147140000, 7148445696}},
{67052240640, {{2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15}, {3, 13, 16, 17, 18, 19, 20, 21, 22}}, {33530112000, 33522128640}},
{321571257120, {{2, 3, 4, 5, 8, 10, 12, 15, 16, 17, 18, 19}, {6, 7, 9, 11, 13, 14, 20, 21, 22, 23}}, {160745472000, 160825785120}},
{1575370944000, {{2, 3, 4, 5, 6, 7, 8, 14, 15, 16, 17, 18, 19}, {9, 10, 11, 12, 13, 20, 21, 22, 23, 24}}, {787652812800, 787718131200}}}

王守恩

  41#估算公式可以这样。——Table[B*Ceiling[n!^(1/B)], {B, 2, 6}, {n, B, 30}]——还是回归《三重内积的最小值》——6#。——谢谢mathe。

公式。{4, 6, 10, 22, 54, 142, 402, 1206, 3810, 12636, 43774, 157824, 590520, 2287072, 9148288, 37719356, 160029670, 697553154, 3119552538, 14295585638, 67052240166, 321571247092, 1575370942646},
B—2。{3, 5, 10, 22, 54, 142, 402, 1206, 3810, 12636, 43776, 157824, 590520, 2287080, 9148320, 37719360, 160029696, 697553280, 3119552640, 14295585696, 67052240640, 321571257120, 1575370944000——A127180

公式。{6, 9, 15, 27, 54, 105, 216, 462, 1026, 2349, 5520, 13305, 32808, 82668, 212559, 557061, 1486461, 4034871, 11131899, 31192017, 88705947, 255872226, 748174926, 2216465271, 6649395807, 20191481970, 62034629790},
B—3。{6, 9, 15, 28, 52, 103, 214, 462, 1026, 2348, 5520, 13304, 32808, 82668,

公式。{12, 16, 24, 36, 60, 100, 176, 320, 592, 1124, 2176, 4280, 8556, 17372, 35784, 74704, 157980, 338180, 732408, 1603924, 3550068, 7938192, 17925232, 40860692, 93992932, 218119604, 510475628},
B—4。{10, 14, 21, 35, 60, 100, 175, 319, 594, 1124, 2174, 4278,

公式。{15, 20, 30, 45, 65, 105, 170, 275, 455, 775, 1330, 2310, 4070, 7250, 13065, 23780, 43720, 81120, 151870, 286760, 545885, 1047355, 2024730, 3942745, 7731750, 15265185},
B—5。{15, 20, 28, 43, 69, 110, 170, 273, 455, 772, 1334, 2308,——40#——B=1,2,3,4,5。

公式。{18, 30, 36, 54, 78, 114, 168, 258, 402, 630, 996, 1602, 2586, 4224, 6960, 11562, 19350, 32628, 55416, 94758, 163092, 282480, 492240, 862800, 1520880}}
B—6。{},

王守恩

B—3。估算公式方向没错。

w(3)=6, {{1}, {2}, {3}}, {1, 2, 3}},
w(4)=9, {{2}, {3}, {4}}, {2, 3, 4}},
w(5)=15, {{2, 3}, {4}, {5}}, {6, 4, 5}},
w(6)=28, {{2, 5}, {3, 4}, {6}}, {10, 12, 6}},
w(7)=52, {{2, 7}, {3, 6}, {4, 5}}, {14, 18, 20}},
w(8)=103, {{2, 3, 6}, {4, 8}, {5, 7}}, {36, 32, 35}},
w(9)=214, {{2, 5, 7}, {3, 4, 6}, {8, 9}}, {70, 72, 72}},
w(10)=462, {{2, 8, 9}, {3, 5, 10}, {4, 6, 7}}, {144, 150, 168}},
w(11)=1026, {{2, 4, 6, 7}, {3, 10, 11}, {5, 8, 9}}, {336, 330, 360}},
w(12)=2348, {{2, 5, 8, 10}, {3, 4, 7, 9}, {6, 11, 12}}, {800, 756, 792}},
w(13)=5520, {{2, 6, 12, 13}, {3, 7, 8, 11}, {4, 5, 9, 10}}, {1872, 1848, 1800}},
w(14)=13304, {{2, 12, 13, 14}, {3, 4, 6, 7, 9}, {5, 8, 10, 11}}, {4368, 4536, 4400}},
w(15)=32808,
w(16)=82668,
w(17)=212560,
w(18)=557064,

得到一串数——6, 9, 15, 28, 52, 103, 214, 462, 1026, 2348, 5520, 13304, 32808, 82668, 212560, 557064, 1486472, 4034874,——天花板了。

这串数——OEIS是不会有的。OEIS有类似的——A61030,31,32,A355189,90,91,92。——可惜没有我们的"细腻"。我们走的是“大路货”。

估算公式可以这样。——Table[3*Ceiling[n!^(1/3)], {n, 3, 30}]——还是回归《三重内积的最小值》——6#。——谢谢mathe。

公式。{6, 9, 15, 27, 54, 105, 216, 462, 1026, 2349, 5520, 13305, 32808, 82668, 212559, 557061, 1486461, 4034871, 11131899, 31192017, 88705947, 255872226, 748174926, 2216465271, 6649395807, 20191481970, 62034629790},
B—3。{6, 9, 15, 28, 52, 103, 214, 462, 1026, 2348, 5520, 13304, 32808, 82668, 212560, 557064, 1486472, 4034874,—— 一直在按这个方向走！前面很多题目都是这样,  由衷的感谢mathe的这个万能公式。

王守恩

B—2。——A127180——有这么个有趣的现象。请看！

0        2
1        2
w(1)=w(0)*1。——正确。

3        5
4        10
w(4)=w(3)*2。——正确。

8        402
9        1206
w(9)=w(8)*3。——正确。

15        2287080
16        9148320
w(16)=w(15)*4。——正确。

24        1575370944000
25        7876854720000
w(25)=w(24)*5。——正确。

35        203304185558353152000
36        1219825113350118912000
w(36)=w(35)*6。——正确。

48        7046677399324048949663605785600
49        49326741795268342647645240499200
w(48)=w(49)*7。——正确。

63        89052980082744902260966290801991589068800000
64        712423840661959218087730326415932712550400000
w(64)=w(63)*8。——正确。

80        535049369857637725242982118321705480362844177301504000000000
81        4815444328718739527186822911399337392982050783362201826099200
w(81)=w(80)*9。——不正确。

98        194184350027320664568896320075343498451557511013681127891101296692620689408000
99        1932109887598985946266001740728895058275207238505714697486173229606240256000000
100        19321098875989859462660017407288950582752072385057146974861732296062402560000000
w(100)=w(99)*10。——正确。

120        5172814674217172090621983541355107064971517693300695790929933747492217192070612084599029760000000000
121        56900961416388892996841818954906177714686694626307653700229271222414389112776732930589327360000000000
w(121)=w(120)*11。——正确。

......

(1)——正确。想想也挺简单——我们称两堆为"u","v"。若 w(n^2-1) 已经找到答案。则w(n^2)—— "u": 放入 n^2,  取出 n。"v": 放入 n 。
也可以这样——若 w(n^2) 已经找到答案。则w(n^2-1)—— "v": 取出 n 。"u": 放入 n,  取出 n^2。 就可以了。

(2)——不正确。？若 w(81) 已经找到答案。则  w(80) —— "v": 取出 9 。"u": 放入 9,  取出 81。w(80)=535049369857637725242982118321705480362844177301504000000000就大了。

(3)——这帖子本来是删了的。想了又想再发上来。主要还是为了”自己“。

王守恩

说得细一点。
         

1. Solve[{A + B == 4815444328718739527186822911399337392982050783362201826099200, A*B == 81!, A/9 == u, B/9 == v, u + v == w, A > B > 0}, {A, B, u, v, w}]

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{A -> 2407722164361916904396003320504471754205926383362201826099200,
  B -> 2407722164356822622790819590894865638776124400000000000000000,
  u -> 267524684929101878266222591167163528245102931484689091788800,
  v -> 267524684928535846976757732321651737641791600000000000000000,
  w -> 535049369857637725242980323488815265886894531484689091788800}——错在那里了？

王守恩

接楼上。63 = 7*9。合不拢。——手工不易——求助。

267524684929101878266222591167163528245102931484689091788800 = 2*4*6*8*9^2*11*12*16*18*22*23*24*26*27*32*33*34*36*42*43*44*46*48*52*54*55*56*58*59*64*66*68*69*72*73*77*78*79*80,
267524684928535846976757732321651737641791600000000000000000 = 3*5*7^2*10*13*14*15*17*19*20*21*25*28*29*30*31*35*37*38*39*40*41*45*47*49*50*51*53*57*60*61*62*65*67*70*71*74*75*76,

或:   40 = 5*8。

267524684929101878266222591167163528245102931484689091788800 = 2*4*6*8^2*9*11*12*16*18*22*23*24*26*27*32*33*34*36*42*43*44*46*48*52*54*55*58*59*63*64*66*68*69*72*73*77*78*79*80,
267524684928535846976757732321651737641791600000000000000000 = 3*5^2*7*10*13*14*15*17*19*20*21*25*28*29*30*31*35*37*38*39*41*45*47*49*50*51*53*56*57*60*61*62*65*67*70*71*74*75*76,

northwolves

王守恩 发表于 2025-10-9 05:16
B—3。估算公式方向没错。

w(3)=6, {{1}, {2}, {3}}, {1, 2, 3}},

1. f[n_] := (a = CubeRoot[n!]; r = 1 + 1/(3 n);  v = Select[Range[Floor[a/r], Ceiling[a*r]], Mod[n!, #] == 0 &];
   
2.   sol = First@SortBy[Select[Subsets[v, {3}], Times @@ # == n! &], Total@# &]; {n, Ceiling[Total@sol, 2], sol});
   
3. Do[Print[f[n]], {n, 15, 20}]

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{15,32808,{10800,10920,11088}}
{16,82668,{27300,27648,27720}}
{17,212560,{70560,70720,71280}}
{18,557064,{184800,185895,186368}}
{19,1486462,{494208,496125,496128}}
{20,4034872,{1343680,1344000,1347192}}

northwolves

从几何均值下手，调整筛选范围r因子即可。但直觉这样得到的三个数字不一定都有解：

{15,32808,{10800,10920,11088}}
{16,82668,{27300,27648,27720}}
{17,212560,{70560,70720,71280}}
{18,557064,{184800,185895,186368}}
{19,1486462,{494208,496125,496128}}
{20,4034872,{1343680,1344000,1347192}}
{21,11131920,{3704400,3706560,3720960}}
{22,31192026,{10388250,10395840,10407936}}
{23,88705948,{29560960,29568000,29576988}}
{24,255872240,{85250880,85299200,85322160}}
{25,748174960,{249318000,249356800,249500160}}
{26,2216465280,{738720000,738840960,738904320}}
{27,6649395840,{2216160000,2216522880,2216712960}}
{28,20191482288,{6729074352,6730407936,6732000000}}
{29,62034630080,{20675655000,20678434920,20680540160}}
{30,192756018766,{64245312000,64253314125,64257392640}}
{31,605520027700,{201819656500,201847852800,201852518400}}
{32,1922406255104,{640760440320,640813814784,640832000000}}
{33,6166184053342,{2055348067500,2055410286592,2055425699250}}
{34,19976042627232,{6658560000000,6658705461408,6658777165824}}

王守恩

估算公式最管用的一串数—— n! 分成2个数的积。2个数的和最小——OEIS还是没有。

1. s = {}; Do[t = n!; r = First@Nearest[Divisors[t], Sqrt[t]]; AppendTo[s, {n, r, t/r, r + t/r}], {n, 24}]; s

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{1, 1, 1, 2},
{2, 1, 2, 3},
{3, 2, 3, 5},
{4, 4, 6, 10},
{5, 10, 12, 22},
{6, 24, 30, 54},
{7, 70, 72, 142},
{8, 192, 210, 402},
{9, 576, 630, 1206},
{10, 1890, 1920, 3810},
{11, 6300, 6336, 12636},
{12, 21600, 22176, 43776},
{13, 78848, 78975, 157823},
{14, 294840, 295680, 590520},
{15, 1143072, 1144000, 2287072},
{16, 4572288, 4576000, 9148288},
{17, 18849600, 18869760, 37719360},
{18, 79968000, 80061696, 160029696},
{19, 348566400, 348986880, 697553280},
{20, 1559376000, 1560176640, 3119552640},
{21, 7147140000, 7148445696, 14295585696},
{22, 33522128640, 33530112000, 67052240640},
{23, 160758097500, 160813154304, 321571251804},
{24, 787652812800, 787718131200, 1575370944000}}
......
C—2。{2, 3, 5, 10, 22, 54, 142, 402, 1206, 3810, 12636, 43776, 157823, 590520, 2287072, 9148288, 37719360, 160029696, 697553280, 3119552640, 14295585696, 67052240640, 321571251804, 1575370944000, 7876854720000,
公式。{2, 4, 6, 10, 22, 54, 142, 402, 1206, 3810, 12636, 43774, 157824, 590520, 2287072, 9148288, 37719356, 160029670, 697553154, 3119552538, 14295585638, 67052240166, 321571247092, 1575370942646, 7876854713230,
C—2。40164235953600, 208699491634480, 1104333907219200, 5947020092055876, 32573170550366496, 181359738136165500, 1025925605497512720, 5893493910706321920, 34364679485752811520, 203304185558353152000,
公式。40164235888492, 208699491618148, 1104333907134458, 5947020092025822, 32573170543389912, 181359738135870972, 1025925605360726984, 5893493910682146958, 34364679485751304814, 203304185558351404344,
C—2。1219825113350118912000, 7419906493003421208000, 45739375486194651955200, 285642308359304874379008, 1806560581046706085708800}
公式。1219825113350108426056, 7419906493002818171382, 45739375486187002015904, 285642308359230589373188, 1806560581046644817271222}

1. C—2。Table[t = n!; r = First@Nearest[Divisors[t], Sqrt[t]]; r + t/r, {n, 40}]

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1. 公式。Table[2 Ceiling[Sqrt[n!]], {n, 40}]

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