有这样一串数(OEIS找不到)

northwolves · 发表于 2024-5-14 13:26:07

数字串(0): 2n个不同整数(最小=0),  n个数的和=n个数的和。我们希望：和是最小的。

n=2k+1:  取1-(4k+2) 即可
n=2k+1:  取1-(4k-1) 即可

{3, 8, 14, 23, 33, 46, 60, 77, 95, 116, 138, 163, 189, 218, 248, 281, 315, 352, 390...}
$a_n=n^2-floor[n/2]$

northwolves · 发表于 2024-5-14 13:44:36

n=11, 82537111381725, {1, 3, 5, 11, 15, 39, 49, 51, 57, 69, 87}, {7, 9, 13, 17, 19, 21, 23, 25, 27, 29, 33}

northwolves · 发表于 2024-5-14 13:47:27

n=12, 5482822398928875, {1, 3, 5, 7, 11, 39, 45, 51, 69, 87, 93, 95}, {9, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33}

northwolves · 发表于 2024-5-14 13:48:54

小一点的解： n=12, 3714170012177625, {1, 3, 5, 7, 25, 33, 39, 51, 57, 63, 69, 87}, {9, 11, 13, 15, 17, 19, 21, 23, 27, 29, 35, 45}

northwolves · 发表于 2024-5-14 13:50:52

n=13, 233992710767190375, {1, 3, 5, 7, 25, 33, 39, 49, 51, 57, 69, 81, 87}, {9, 11, 13, 15, 17, 19, 21, 23, 27, 29, 35, 45, 63}

mathe · 发表于 2024-5-14 17:14:02

N=2
15<1>
{ 1 3 5 15 }<1>

N=3
225<1>
{ 1 3 5 9 15 25 }<1>

N=4
2835<1>
{ 1 3 5 7 9 15 21 27 }<3>

N=5
51975<1>
{ 1 3 5 7 9 11 15 21 25 33 }<4>

N=6
1216215<1>
{ 1 3 5 7 9 11 13 15 21 27 33 39 }<11>

N=7
34459425<1>
{ 1 3 5 7 9 11 13 15 17 21 25 33 39 51 }<14>

N=8
1178512335<1>
{ 1 3 5 7 9 11 13 15 17 19 21 27 33 39 51 57 }<41>

N=9
41247931725<1>
{ 1 3 5 7 9 11 13 15 17 19 21 25 27 33 39 49 51 57 }<37>

N=10
1856156927625<1>
{ 1 3 5 7 9 11 13 15 17 19 21 25 27 33 35 39 45 51 57 63 }<207>

N=11
82537111381725<1>
{ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 33 39 49 51 57 69 87 }<138>

N=12
3714170012177625<1>
{ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 33 35 39 45 51 57 63 69 87 }<784>

N=13
233992710767190375<1>
{ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 33 35 39 45 49 51 57 63 69 81 87 }<1622>

N=14
12089623389638169375<1>
{ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 39 45 49 51 57 63 69 75 87 93 }<3446>

N=15
805168917749902080375<1>
{ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 45 49 51 57 63 69 81 87 93 111 }<6248>

N=16
55019876046243308825625<1>
{ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 45 49 51 57 63 69 75 87 93 111 123 }<13151>

王守恩 · 发表于 2024-5-14 19:07:59

不知道哪里卡住了, 出不来, 再来一串试试？谢谢各位大侠！

数字串(2): 2n个不同正整数(只能是2,3,5,7的倍数), n个数的积=n个数的积。我们希望：积是最小的。

N=2
6
{ 2 3 }{ 1 6 }
N=3
36
{ 2 3 6 }{ 1 4 9 }
N=4
240
{ 2 4 5 6 }{ 1 3 8 10 }
N=5
2520
{ 3 4 5 6 7 }{ 1 2 9 10 14 }
N=6
30240
{ 2 5 6 7 8 9 }{ 1 3 4 12 14 15 }

mathe · 发表于 2024-5-14 21:23:14

Result for A354697

N=2
12<1>
{ 2 3 4 6 }<1>

N=3
120<4>
{ 2 3 5 6 8 10 }<1>
{ 2 3 4 5 10 12 }<1>
{ 2 3 4 5 8 15 }<1>
{ 2 3 4 5 6 20 }<1>

N=4
720<1>
{ 2 3 4 5 6 8 9 10 }<2>

N=5
10080<7>
{ 2 3 4 5 7 8 9 10 12 14 }<4>
{ 2 3 4 5 6 7 9 10 14 16 }<2>
{ 2 3 4 5 6 7 8 12 14 15 }<5>
{ 2 3 4 5 6 7 8 10 14 18 }<3>
{ 2 3 4 5 6 7 8 10 12 21 }<5>
{ 2 3 4 5 6 7 8 9 14 20 }<3>
{ 2 3 4 5 6 7 8 9 10 28 }<3>

N=6
110880<1>
{ 2 3 4 5 6 7 8 9 10 11 14 22 }<7>

N=7
1814400<1>
{ 2 3 4 5 6 7 8 9 10 12 14 15 18 20 }<31>

N=8
26611200<1>
{ 2 3 4 5 6 7 8 9 10 11 12 14 15 16 20 22 }<55>

N=9
518918400<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 18 20 22 26 }<114>

N=10
10378368000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 22 25 26 }<237>

N=11
261534873600<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 21 22 26 27 28 }<695>

N=12
5928123801600<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 24 26 28 34 }<1666>

N=13
168951528345600<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 26 27 28 34 38 }<2646>

N=14
4505374089216000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26 28 32 34 38 }<6928>

N=15
152056375511040000<2>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26 27 30 34 35 36 38 }<21512>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 25 26 27 28 30 34 38 45 }<21474>

N=16
4663062182338560000<2>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 30 32 34 35 38 46 }<37227>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 30 34 38 40 46 }<41871>

N=17
167870238564188160000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 30 32 34 36 38 45 46 }<126721>

N=18
6463004184721244160000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 30 33 34 35 36 38 42 44 46 }<375348>

N=19
249902828475888107520000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 32 33 34 35 38 42 44 46 58 }<667321>

N=20
10495918795987300515840000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 32 33 34 35 36 38 42 44 46 49 58 }<1831927>

N=21
449825091256598593536000000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 32 33 34 35 36 38 40 42 44 45 46 50 58 }<7130833>

N=22
19522408960536378959462400000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 38 40 42 44 45 46 49 58 62 }<12067929>

N=23
870141656526764319336038400000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 38 39 40 42 44 45 46 48 52 58 62 }<42973699>

N=24
45682436967655126765142016000000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 38 39 40 42 44 45 46 49 50 52 54 58 62 }<105786888>

N=25
2253666890404319587080339456000000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 42 44 45 46 48 49 50 52 58 62 74 }<218943019>

N=26
137924413692744358729316774707200000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 42 44 45 46 48 49 51 52 54 58 62 68 74 }<646950177>

N=27
7761628770552476657904689086464000000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 44 45 46 48 49 50 52 56 58 62 63 74 82 }<1476274502>

N=28
452392076912201496632159021039616000000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 44 45 46 48 49 50 51 52 54 58 62 64 68 74 82 }<3846678717>

N=29
28500700845468694287826018325495808000000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 44 45 46 48 49 50 51 52 54 56 58 62 63 68 72 74 82 }<14320262729>

N=30
1567538546500778185830431007902269440000000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 44 45 46 48 49 50 51 52 54 55 56 58 60 62 63 66 68 74 82 }<46445678648>

N=31
107846651999253539185133653343676137472000000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 48 49 51 52 54 55 56 58 60 62 63 64 66 68 74 82 86 }<91771247330>

N=32
6335990804956145427126602133940973076480000000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 54 55 56 58 60 62 63 66 68 74 82 86 94 }<182567269925>

N=33
456191337956842470753115353643750061506560000000<1>
{ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 54 55 56 58 60 62 63 64 66 68 74 81 82 86 94 }<546927901360>

northwolves · 发表于 2024-5-14 22:04:39

本帖最后由 northwolves 于 2024-5-14 22:48 编辑

王守恩发表于 2024-5-14 19:07
不知道哪里卡住了, 出不来, 再来一串试试？谢谢各位大侠！

数字串(2): 2n个不同正整数(只能是2,3,5,7的倍 ...

n=2,6,{1,6},{2,3}

n=3,36,{1,4,9},{2,3,6}

n=4,240,{1,3,8,10},{2,4,5,6}

n=5,2520,{1,2,9,10,14},{3,4,5,6,7}

n=6,30240,{1,2,6,10,12,21},{3,4,5,7,8,9}

n=7,483840,{1,2,5,9,14,16,24},{3,4,6,7,8,10,12}

n=8,7257600,{1,2,3,14,15,16,18,20},{4,5,6,7,8,9,10,12}

n=9,203212800,{1,2,3,8,15,20,21,24,28},{4,5,6,7,9,10,12,14,16}

n=10,4572288000,{1,2,3,6,16,20,21,25,27,28},{4,5,7,8,9,10,12,14,15,18}

n=11,137168640000,{1,2,3,4,16,21,24,25,27,30,35},{5,6,7,8,9,10,12,14,15,18,20}

n=12,4389396480000,{1,2,3,4,12,21,25,27,28,30,32,40},{5,6,7,8,9,10,14,15,16,18,20,24}

n=13,184354652160000,{1,2,3,4,6,25,27,28,30,32,35,36,56},{5,7,8,9,10,12,14,15,16,18,20,21,24}

n=14,7742895390720000,{5,6,7,8,10,12,14,15,16,18,20,27,28,42},{1,2,3,4,9,21,24,25,30,32,35,36,48,49}

n=15,331838373888000000,{5,6,7,8,10,12,14,15,16,18,20,27,28,36,50},{1,2,3,4,9,21,24,25,30,32,35,40,42,45,48}

n=16,18582948937728000000,{5,6,7,8,10,12,14,15,16,18,20,27,28,42,48,50},{1,2,3,4,9,21,24,25,30,32,35,36,40,45,49,64}

n=17,1170725783076864000000,{5,6,7,8,10,12,14,15,16,18,20,27,28,42,48,50,63},{1,2,3,4,9,21,24,25,30,32,35,36,40,45,49,56,72}

n=18,70243546984611840000000,{5,6,7,8,10,12,14,15,16,18,20,27,28,42,48,50,60,63},{1,2,3,4,9,21,24,25,30,32,35,36,40,45,49,54,56,80}

n=19,5619483758768947200000000,{5,6,7,8,10,12,14,15,16,18,20,27,28,42,48,50,60,63,80},{1,2,3,4,9,21,24,25,30,32,35,36,40,45,49,56,64,72,75}

王守恩 · 发表于 2024-5-15 08:00:40

自我巩固巩固。目的很明确: 搞个无限模式出来！

数字串(0): 2n个不同整数(最小=0), n个数的和=n个数的和。我们希望：和是最小的。A014848好像没说有这个功能。
{3, 8, 14, 23, 33, 46, 60, 77, 95, 116, 138, 163, 189, 218, 248, 281, 315, 352, 390, 431, 473, 518, 564, 613, 663, 716, 770, 827,
885, 946, 1008, 1073, 1139, 1208, 1278, 1351, 1425, 1502, 1580, 1661, 1743, 1828, 1914, 2003, 2093, 2186, 2280, 2377, 2475}

Table[Ceiling[((2 n - 1) n)/2], {n, 2, 50}]

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数字串(1): 2n个不同整数(最小=1), n个数的和=n个数的和。我们希望：和是最小的。A171769好像没说有这个功能。
{5, 11, 18, 28, 39, 53, 68, 86, 105, 127, 150, 176, 203, 233, 264, 298, 333, 371, 410, 452, 495, 541, 588, 638, 689, 743, 798, 856,
915, 977, 1040, 1106, 1173, 1243, 1314, 1388, 1463, 1541, 1620, 1702, 1785, 1871, 1958, 2048, 2139, 2233, 2328, 2426, 2525}

Table[Ceiling[((2 n + 1) n)/2], {n, 2, 50}]

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数字串(2): 2n个不同整数(最小=2), n个数的和=n个数的和。我们希望：和是最小的。A156859好像没说有这个功能。
{7, 14, 22, 33, 45, 60, 76, 95, 115, 138, 162, 189, 217, 248, 280, 315, 351, 390, 430, 473, 517, 564, 612, 663, 715, 770, 826, 885,
945, 1008, 1072, 1139, 1207, 1278, 1350, 1425, 1501, 1580, 1660, 1743, 1827, 1914, 2002, 2093, 2185, 2280, 2376, 2475, 2575}

Table[Ceiling[((2 n + 3) n)/2], {n, 2, 50}]

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数字串(3): 2n个不同整数(最小=3), n个数的和=n个数的和。我们希望：和是最小的。OEIS没有这串数。
{9, 17, 26, 38, 51, 67, 84, 104, 125, 149, 174, 202, 231, 263, 296, 332, 369, 409, 450, 494, 539, 587, 636, 688, 741, 797, 854, 914,
975, 1039, 1104, 1172, 1241, 1313, 1386, 1462, 1539, 1619, 1700, 1784, 1869, 1957, 2046, 2138, 2231, 2327, 2424, 2524, 2625}

Table[Ceiling[((2 n + 5) n)/2], {n, 2, 50}]

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数字串(4): 2n个不同整数(最小=4), n个数的和=n个数的和。我们希望：和是最小的。OEIS没有这串数。
{11, 20, 30, 43, 57, 74, 92, 113, 135, 160, 186, 215, 245, 278, 312, 349, 387, 428, 470, 515, 561, 610, 660, 713, 767, 824, 882, 943,
1005, 1070, 1136, 1205, 1275, 1348, 1422, 1499, 1577, 1658, 1740, 1825, 1911, 2000, 2090, 2183, 2277, 2374, 2472, 2573, 2675}

Table[Ceiling[((2 n + 7) n)/2], {n, 2, 50}]

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数字串(5): 2n个不同整数(最小=5), n个数的和=n个数的和。我们希望：和是最小的。OEIS没有这串数。
{13, 23, 34, 48, 63, 81, 100, 122, 145, 171, 198, 228, 259, 293, 328, 366, 405, 447, 490, 536, 583, 633, 684, 738, 793, 851, 910, 972,
1035, 1101, 1168, 1238, 1309, 1383, 1458, 1536, 1615, 1697, 1780, 1866, 1953, 2043, 2134, 2228, 2323, 2421, 2520, 2622, 2725}

Table[Ceiling[((2 n + 9) n)/2], {n, 2, 50}]

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如果限制底数是某个素数: 2n个不同正整数, n个数的积=n个数的积。我们希望：积是最小的。指数就是这些数字串(无限模式)。

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[求助] 有这样一串数(OEIS找不到)

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