\(a(n)=x,x^{n+1}=y^n+z^n,x≠y\)
\(a(1)=3,3^2=3^1+6^1\)——这个x不是最小的。
\(a(2)=5,5^3=5^2+10^2\)——这个x是最小的。
\(a(3)=9,9^4=9^3+18^3\)——这个x是最小的。
\(a(4)=17,17^5=17^4+34^4\)——这个x是最小的。
\(a(5)=33,33^6=33^5+66^5\)——这个x是最小的。
\(a(6)=65,65^7=65^6+130^6\)——这个x是最小的。
\(a(7)=129,129^8=129^7+258^7\)——这个x是最小的。
\(a(8)=257,257^9=257^8+514^8\)——这个x是最小的。
\(a(9)=513,513^{10}=513^9+1026^9\)——这个x是最小的。
3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193,16385, 32769, 65537, 131073, 262145, 524289, 1048577——a(n) = 2^n + 1——A000051——还没有这样的条文。
\(a(n)=x,x^{n+1}=y^n+z^n,x≠y≠z\)
\(a(1)=5,5^2=10^1+15^1\)——这个x不是最小的。
\(a(2)=13,13^3=26^2+39^2\)——这个x不是最小的。
\(a(3)=35,35^4=70^3+105^3\)——这个x是最小的。
\(a(4)=97,97^5=194^4+291^4\)——这个x是最小的。
\(a(5)=275,275^6=550^5+825^5\)——这个x是最小的。
\(a(6)=793,793^7=1586^6+2379^6\)——这个x是最小的。
\(a(7)=2315,2315^8=4630^7+6945^7\)——这个x是最小的。
\(a(8)=6817,6817^9=13634^8+20451^8\)——这个x是最小的。
\(a(9)=20195,20195^{10}=40390^9+605855^9\)——这个x是最小的。
a(n) = 2^n + 3^n——A007689——这个x是最小的。这句话能成立的话——这个x是最小的。这句话应该成为——A007689——当家条文。
5, 13, 35, 97, 275, 793, 2315, 6817, 20195, 60073, 179195, 535537, 1602515, 4799353, 14381675, 43112257, 129271235, 387682633, 1162785755, 3487832977, 10462450355, 31385253913, 94151567435, 282446313697,
A300565——Numbers z such that there is a solution to x^3 + y^4 = z^5 with x, y, z >= 1.
z=32, 250, 1944, 2744, 3888, 19208, 27648, 55296, 59049, 59582, 81000, 82944, 131072, 135000, 185193, 200000,
# A300565 (b-file synthesized from sequence entry)——后面还有大把地皮空置——我来把 y,x 补上。
1 32{{y -> 64, x -> 256}}
2 250{{y -> 625, x -> 9375}}
3 1944{{y -> 11664, x -> 209952}}
4 2744{{y -> 19208, x -> 268912}}
5 3888{{y -> 23328, x -> 839808}}
6 19208{{y -> 134456, x -> 13176688}}
7 27648{{y -> 331776, x -> 15925248}}
8 55296{{y -> 331776, x -> 79626240}}
9 59049{{y -> 531441, x -> 86093442}}
10 59582{{y -> 923521, x -> 28629151}}
11 81000{{y -> 1215000,x -> 109350000}}
12 82944{{y -> 995328,x -> 143327232}}
13 131072{{y -> 2097152, x -> 268435456}}
14 135000{{y -> 2025000, x -> 303750000}}
15 185193{{y -> 3518667, x -> 401128038}}
16 200000{{y -> 4000000, x -> 400000000}}
对比A303265——才知道上面这些数据来之不易。
已知直角四面体六条棱之和=n(n=9, 10, 11, 12, 13, ...), 则四面体的最大体积=
{0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 20, 22, 24, 26, 28, 30, 33, 35, 37, 40, 43, 46, 49, 52, 55, 58, 62, 65, 69, 73, 77, 81, 86, 90, 95, 100, 105, 110, 115, 120, 126, 132, 138, 144, 150, 157, 164}
Table - 1)*n/3)^3/6], {n, 9, 72}]
northwolves 发表于 2024-12-9 09:19
这个代码更直观些：
A293173——1, 2, 5, 13, 17, 34, 41, 85, 89, 233, 305, 386, 481, 610, 937, 1213, 1597, 1762, 3653, 4181, 5473, 6850, 8077, 8321, 8857, 10946, 21029, 21506, 28657, 40097, 47125, 75025, 75725, 98209,
Flatten@Table, {y, 123}, {z, 123}]
{x -> 1, x -> 1, x -> 1, x -> 2, x -> 2, x -> 5, x -> 5, x -> 13, x -> 13, x -> 34, x -> 34, x -> 89, x -> 1, x -> 1, x -> 1, x -> 17, x -> 1, x -> 2, x -> 1, x -> 41, x -> 2, x -> 85, x -> 2, x -> 5,
x -> 2, x -> 481, x -> 1, x -> 17, x -> 1, x -> 386, x -> 1, x -> 41, x -> 1, x -> 937, x -> 5, x -> 13, x -> 2, x -> 85, x -> 1, x -> 386, x -> 13, x -> 34, x -> 1, x -> 937, x -> 2, x -> 481, x -> 34, x -> 89}
——我还是不知道怎么把"x"去掉,然后合并一下——这次与上次不一样(564#,565#,566#)——上次只有"1个"(y), 这次有"2个"(y,z)。
王守恩 发表于 2025-7-27 05:20
A293173——1, 2, 5, 13, 17, 34, 41, 85, 89, 233, 305, 386, 481, 610, 937, 1213, 1597, 1762, 3653, 4 ...
s = Values@
Solve[{5x(y*z - x) == y^2 + z^2, y <= z <= 50}, {x, y, z},
PositiveIntegers]
A345731——Additive bases: a(n) is the least integer such that there is an n-element set of integers between 0 and a(n), the sums of pairs (of distinct elements) of which are distinct.
A345731——1, 2, 4, 7, 12, 18, 24, 34, 45, 57, 71, 86, 105, 126, 150, 171,——Table of n, a(n) for n=2..17.——{0, 1, 2, 3, ..., a(n)}没有等和对的最大子集。
a(1)=0,0,
a(2)=1,0, 1,
a(3)=2,0, 1, 2,
a(4)=4,0, 2, 3, 4,
a(5)=7,0, 3, 5, 6, 7,
a(6)=12,0, 1, 2, 6, 9, 12,
a(7)=18,0, 1, 2, 4, 8, 13, 18,
a(8)=24,0, 1, 2, 4, 8, 14, 19, 24,
a(9)=34,0, 1, 2, 4, 8, 15, 24, 29, 34,
a(10)=45,0, 1, 7, 10, 13, 21, 26, 41, 43, 45,
a(11)=57,0, 1, 5, 9, 17, 31, 33, 44, 51, 54, 57,
a(12)=71,0, 1, 2,7, 12 22, 37, 40, 54, 63, 67, 71,
a(13)=86,0, 1, 11, 17, 21, 34, 42, 57, 60, 72, 79, 84, 86,——72,79,86可以成等差数列。
a(14)=105,0, 1, 6, 14, 27, 44 54 66, 69, 85, 94, 101, 103, 105,
a(15)=126,0, 8, 13, 16, 19, 31, 46, 76, 83, 90, 100, 104, 124, 125, 126,
a(16)=148,0, 3, 5, 6, 32, 49, 59, 68, 93, 106, 118, 126, 130, 134, 141, 148,
a(17)=171,
挺好的三串数——A(n),B(n),C(n)——谢谢 northwolves ！！！
Table + Sin + Sin)^(2 n - 1) ==A*Sin^(2 n - 1) + B*Sin^(2 n - 1) + Sin^(2 n - 1)}, {A, B}, PositiveIntegers], {n, 9}]
n=1, A(1)=1, B(1)=1,
n=2, A(2)=46, B(2)=10,
n=3, A(3)=1156, B(3)=76,
n=4, A(4)=26440, B(4)=568,
n=5, A(5)=594352, B(5)=4240,
n=6, A(6)=13318240, B(6)=31648,
n=7, A(7)=298263616, B(7)=236224,
n=8, A(8)=6678960256, B(8)=1763200,
n=9, A(9)=149557916416, B(9)=13160704,
B(n)=1, 10, 76, 568, 4240, 31648, 236224, 1763200, 13160704, 98232832, 733219840, 5472827392, 40849739776, 304906608640, 2275853910016, ——A107903——Generalized NSW numbers.——条文没我们的有意义。
B(n)=A107903(n)=Floor[(1+Sqrt{3})^(2n-1)/2]
A(n)=1, 46, 1156, 26440, 594352, 13318240, 298263616, 6678960256, 149557916416, 3348948866560, 74990693573632, 1679214509639680, 37601483354976256,——OEIE就没有了。
Table[(((1 + Sqrt) (2 - Sqrt)^n - (1 - Sqrt) (2 + Sqrt)^n) 6^n)/(4 Sqrt) - 2^(2 n - 1), {n, 20}] // FullSimplify
固定n, 在A(n), B(n)有解的前提下,C(n)表示可以取到的最大值。
Table + Sin + Sin)^(2 n - 1) ==A*Sin^(2 n - 1) + B*Sin^(2 n - 1) + C*Sin^(2 n - 1)}, {A, B, C}, PositiveIntegers], {n, 3}]
C(n)=1, 6, 37, 207, 1161, 6504, 36410, 203826, 1141037, 6387614, 35758350,——电脑罢工了——好不容易搞了这么几个————连我也找不到通项公式。
Table)^(2 n - 1)/4^n], {n, 20}]——谢谢 northwolves ！！！
三角数A=n(n+1)/2的数字和，
数字和=1，A={1, 4,
数字和=3, A={2, 6, 15, 20, 24, 141, 200, 2000, 20000, 200000, 2000000,
数字和=6, A={3, 5, 14, 21, 66, 77, 201, 473, 2001, 15620, 20001, 200001, 2000001,
数字和=9, A={8, 9, 17, 18, 26, 35, 45, 53, 63, 80, 81, 89, 126, 144, 161, 162, 179, 206, 215, 224, 449, 458, 477, 666, 800, 801, 1421, 1575, 1620, 1673, 2006, 2015, 2195, 2835, 4473, 4733, 6326, 8000, 8001, 8126, 14184, 14220,
数字和=12, A={11, 29, 33, 38, 42, 47, 51, 60, 65, 69, 78, 101, 110, 119, 146, 150, 155, 159, 164, 173, 195, 204, 245, 249, 258, 317, 326, 375, 402, 447, 510, 533, 600, 632, 663, 681, 722, 1001, 1095, 1122, 1266, 1415, 1424, 1428,
数字和=15, A={12, 23, 30, 32, 39, 41, 48, 50, 57, 59, 68, 75, 84, 86, 93, 95, 102, 104, 111, 113, 120, 122, 149, 156, 158, 167, 174, 185, 203, 210, 219, 221, 228, 237, 246, 255, 257, 266, 273, 284, 293, 300, 318, 320, 374, 401, 453,
数字和=18, A={27, 36, 44, 54, 62, 71, 72, 90, 98, 99, 117, 134, 135, 143, 152, 153, 170, 171, 180, 197, 198, 207, 216, 225, 233, 242, 251, 260, 261, 269, 279, 287, 288, 297, 323, 324, 333, 341, 350, 359, 377, 378, 395, 404, 405, 440,
数字和=21, A={56, 74, 83, 87, 92, 96, 105, 114, 123, 128, 137, 168, 177, 182, 186, 191, 209, 213, 218, 222, 240, 254, 263, 272, 276, 281, 285, 290, 294, 299, 303, 321, 330, 335, 339, 344, 348, 353, 357, 362, 366, 371, 380, 384, 393,
把第一个数取出来(我的电脑只能出来这么几个)——OEIS没有这串数。
{1, 2, 3, 8, 11, 12, 27, 56, 129, 107, 132, 309, 368, 627, 968, 1332, 3129, 3434, 5291, 8831, 13332, 18972, 28248, 37067, 77067, 107516, 140547, 278172, 368507,
Sum[((2 n - 3)!!/(2 n)!!)^2, {n, 0, Infinity}]
4/Pi
Table, {a, 3}]
{Hypergeometric2F1[-(1/2), -(1/2), 1, 2/3], Hypergeometric2F1[-(1/2), -(1/2), 1, 3/4], Hypergeometric2F1[-(1/2), -(1/2), 1, 4/5]}
Table[(Sum[(2 n - 3)!!/((2 n)!! k^n), {n, 0, Infinity}])^2, {k, 2, 23}]
{1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10, 10/11, 11/12, 12/13, 13/14, 14/15, 15/16, 16/17, 17/18, 18/19, 19/20, 20/21, 21/22, 22/23}
Table[(Sum[(2 n - 1)!!/((2 n)!! k^n), {n, 0, Infinity}])^2, {k, 2, 25}]
{2, 3/2, 4/3, 5/4, 6/5, 7/6, 8/7, 9/8, 10/9, 11/10, 12/11, 13/12, 14/13, 15/14, 16/15, 17/16, 18/17, 19/18, 20/19, 21/20, 22/21, 23/22, 24/23, 25/24}
Table[(Sum[(2 n - 3)!!/((2 n)!! ((k + 1)/k)^n), {n, 0, Infinity}])^2, {k, 29}]
{1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20, 1/21, 1/22, 1/23, 1/24, 1/25, 1/26, 1/27, 1/28, 1/29, 1/30}
Table[(Sum[(2 n - 1)!!/((2 n)!! ((k + 1)/k)^n), {n, 0, Infinity}])^2, {k, 49}]
{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}