求{1, 2, ..., 100}没有等和对的最大子集

王守恩

northwolves 发表于 2025-7-21 23:37
Table 改为 Do 即可：

搬运一下——用了4个小时。
Do[K={0, a1, a2, a3, a4, a5, a6, a7, a8, 45};S=Total/@Subsets[K,{2}];If[Length[S]==Length[Union[S]],Print[{Length[S],K}]],{a1, 37},{a2, a1+1, 38},{a3, a2+1, 39},{a4, a3+1, 40},{a5, a4+1, 41},{a6, a5+1, 42},{a7, a6+1, 43},{a8, a7+1, 44}]
{45,{0,1,7,10,13,21,26,41,43,45}}
{45,{0,2,4,19,24,32,35,38,44,45}}

王守恩

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,

王守恩

A={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ..., k}
我们在A里取n个不同数,满足两两之和均不相同。譬如：
a(2)=1, {0,1}, 1+0.
a(3)=2, {0,1,2}, 1+0; 2+0,2+1.
a(4)=4, {0,1,2,4}, 1+0; 2+0,2+1; 4+0,4+1,4+2.
a(5)=7, {0,1,2,4,7}, 1+0; 2+0,2+1; 4+0,4+1,4+2; 7+0,7+1,7+2,7+4.
a(6)=12, {0,1,2,6,9,12}, 1+0; 2+0,2+1; 6+0,6+1,6+2; 9+0,9+1,9+2,9+6; 12+0,12+1,12+2,12+6,12+9.
a(7)=18, {0,1,2,4,8,13,18}, 1+0; 2+0,2+1; 4+0,4+1,4+2; 8+0,8+1,8+2,8+4; 13+0,13+1,13+2,13+4,13+8; 18+0,18+1,18+2,18+4,18+8,18+13.
a(8)=24, {0,1,2,4,8,14,19,24}, 1+0; 2+0,2+1; 4+0,4+1,4+2; 8+0,8+1,8+2,8+4; 14+0,14+1,14+2,14+4,14+8; 19+0,19+1,19+2,19+4,19+8,19+14; 24+0,24+1,24+2,24+4,24+8,24+14,24+19.
a(9)=34, {0,1,2,4,8,15,24,29,34}, 1+0; 2+0,2+1; 4+0,4+1,4+2; 8+0,8+1,8+2,8+4; 15+0,15+1,15+2,15+4,15+8; 24+0,24+1,24+2,24+4,24+8,24+15; 29+0,29+1,29+2,29+4,29+8,29+15,29+24; 34+0,34+1,34+2,34+4,34+8,34+15,34+24,34+29.
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},
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,——？？？
a(18)=???,——？？？
a(19)=???,——？？？

A345731——1, 2, 4, 7, 12, 18, 24, 34, 45, 57, 71, 86, 105, 126, 150, 171,——目前为止也就这16项——没有等和对的最大子集。
A003022——0, 1, 3, 6, 11, 17, 25, 34, 44, 55, 72, 85, 106, 127, 151, 177, 199, 216, 246, 283, 333, 356, 372, 425, 480, 492, 553, 585,——目前为止也就这28项——没有等差对的最大子集。
A={0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ..., k}
我们在A里取n个不同数,满足两两之差均不相同。譬如：
a(2)=1, {0,1}, 1-0.
a(3)=3, {0,1,3}, 1-0; 3-0,3-1.
a(4)=6, {0,1,4,6}, 1-0; 4-0,4-1; 6-0,6-1,6-4.
a(5)=11, {0,1,4,9,11}, 1-0; 4-0,4-1; 9-0,9-1,9-4; 11-0,11-1,11-4,11-9.
a(6)=17, {0,1,4,10,12,17}, 1-0; 4-0,4-1; 10-0,10-1,10-4; 12-0,12-1,12-4,12-10; 17-0,17-1,17-4,17-10,17-12.
a(7)=25, {0,1,4,10,18,23,25}, 1-0; 4-0,4-1; 10-0,10-1,10-4; 10-0,18-1,18-4,18-10; 23-0,23-1,23-4,23-10,23-18; 25-0,25-1,25-4,25-10,25-18,25-23.
a(8)=34, {0,1,4,9,15,22,32,34}, 1-0; 4-0,4-1; 9-0,9-1,9-4; 15-0,15-1,15-4,15-9; 22-0,22-1,22-4,22-9,22-15; 32-0,32-1,32-4,32-9,32-15,32-22; 34-0,34-1,34-4,34-9,34-15,34-22,34-32.
a(9)=44, {0,1,5,12,25,27,35,41,44}, 1-0; 5-0,5-1; 12-0,12-1,12-5; 25-0,25-1,25-5,25-12; 27-0,27-1,27-5,27-12,27-25; 35-0,35-1,35-5,35-12,35-25,35-27; 41-0,41-1,41-5,41-12,41-25,41-27,41-35; 44-0,44-1,44-5,44-12,44-25,44-27,44-35,44-41.
a(10)=55, {0, 1, 6, 10, 23, 26, 34, 41, 53, 55},
a(11)=72, {0, 1, 4, 13, 28, 33, 47, 54, 64, 70, 72},
a(12)=85, {0, 2, 6, 24, 29, 40, 43, 55, 68, 75, 76, 85},
a(13)=106,——？？？
a(14)=127,——？？？
a(15)=151,——？？？
a(16)=177,——？？？
a(17)=199,——？？？
a(18)=216,——？？？
a(18)=246,——？？？
a(19)=283,——？？？

？——这些？？？我们真还动不了了吗？？？

王守恩

A345731——1, 2, 4, 7, 12, 18, 24, 34, 45, 57, 71, 86, 105, 126, 148 —— Zhao Hui Du, Jul 27 2025。