从下列数串中取两数（允许重复）相加，可得到所有正偶数，这样的数串还可更“稀”吗？

王守恩

有这样一串数：

1, 3, 7, 9, 19, 21, 25, 27, 55, 57, 61, 63, 73, 75, 79, 81, 163, 165,
169, 171, 181, 183, 187, 189, 217, 219, 223, 225, 235, 237, 241,
243, 487, 489, 493, 495, 505, 507, 511, 513, 541, 543, 547, 549,
559, 561, 565, 567, 649, 651, 655, 657, 667, 669, 673, 675, 703,
705, 709, 711, 721, 723, 727, 729, 1459, ..........

1，通项公式是这样：

    Table[2 FromDigits[IntegerDigits[n, 2], 3] + 1, {n, 0, 64}]

2，在这串数中，每次取 2 个数(允许重复取)相加，
    和可以是所有偶数(2, 4, 6, 8, 10, 12, ..........)

3，这样的数字串还有吗？

4，这样的数字串还会有“稀”一点的吗？

谢谢大家！

王守恩

2个奇数(>1)相加, 和跑遍6, 8, 10, 12, 14, 16, 18, 20, ...。那是怎样的一串数？

a(1)=6。  1个数可以算6。           1个数={3}
a(2)=10。2个数可以算6,  8, 10。 2个数={3, 5}
a(3)=14。3个数可以算6——14。3个数={3, 5, 7}
a(4)=20。4个数可以算6——20。4个数={3, 5, 7, 13}
a(5)=26。5个数可以算6——26。5个数={3, 5, 7, 13, 19}
a(6)=34。6个数可以算6——34。6个数={3, 5, 7, 9, 17, 25}
a(7)=42。7个数可以算6——42。7个数={3, 5, 7, 9, 17, 25, 33}
a(8)=52。8个数可以算6——52。8个数={3, 5, 7, 9, 11, 21, 31, 41}
a(9)=62。9个数可以算6——62。9个数={3, 5, 7, 9, 11, 21, 31, 41, 51}
......

得到一串数——{6, 10, 14, 20, 26, 34, 42, 52, 62, 74, 86, 100, 114, 130, 146, 164, 182, 202, 222, 244, 266, 290, 314, 340, 366, 394, 422, 452, ——OEIS有这串数——没有这种"猜想"。

各位！——我们奢望这串数能“稀一点”。——可以“稀一点”吗？  谢谢各位！！！

王守恩

2个奇数(>1)相加(不允许重复), 和跑遍 8, 10, 12, 14, 16, 18, 20, ...。a(n)是最大的那个和。
{2, {8, {3, 5}}},
{3, {12, {3, 5, 7}}},
{4, {18, {3, 5, 7, 11}}},
{5, {24, {3, 5, 7, 11, 17}}},
{6, {32, {3, 5, 7, 11, 17, 23}}},
{7, {40, {3, 5, 7, 11, 17, 23, 31}}},
{8, {50, {3, 5, 7, 11, 17, 23, 31, 39}}},
{9, {60, {3, 5, 7, 11, 17, 23, 31, 39, 49}}},
{10, {72, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59}}},
{11, {84, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71}}},
{12, {98, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83}}},
{13, {112, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97}}},
{14, {128, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111}}},
{15, {144, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127}}},
{16, {162, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143}}},
{17, {180, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161}}},
{18, {200, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161,179}}},
{19, {220, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199}}},
{20, {242, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219}}},
{21, {264, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241}}},
{22, {288, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263}}},
{23, {312, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287}}},
{24, {338, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311}}},
{25, {364, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337}}},
{26, {392, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363}}},
{27, {420, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391}}},
{28, {450, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419}}}}
{29, {480, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449}}},
{30, {512, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449, 479}}},
{31, {544, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449, 479, 511}}},
{32, {578, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449, 479, 511, 543}}},
{33, {612, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449, 479, 511, 543, 577}}},
{34, {648, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449, 479, 511, 543, 577, 611}}},
{35, {684, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449, 479, 511, 543, 577, 611, 647}}},
{36, {722, {3, 5, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449, 479, 511, 543, 577, 611, 647, 683}}}}
......

王守恩

2#,3#都是错的！——下面的比2#,3#"稀"一些。我不敢保证这是最"稀"的。

2个奇数(>1)相加(允许重复), 和跑遍6, 8, 10, 12, 14, 16, 18, 20, ...。a(n)是最大的那个和。

{1, 6, {3}},
{2, 10, {3, 5}},
{3, 14, {3, 5, 7}},
{4, 22, {3, 5, 9, 11}},
{5, 30, {3, 5, 9, 13, 15}},
{6, 38, {3, 5, 9, 13, 17, 19}},
{7, 46, {3, 5, 7, 13, 19, 21, 23}},
{8, 58, {3, 5, 7, 13, 19, 25, 27, 29}},
{9, 70, {3, 5, 7, 13, 19, 25, 31, 33, 35}},
{10, 86, {3, 5, 9, 11, 21, 25, 35, 37, 41, 43}},
{11, 98, {3, 5, 7, 9, 17, 25, 33, 41, 45, 47, 51}},
{12, 114, {3, 5, 7, 9, 17, 25, 33, 41, 49, 53, 55, 59}},
{13, 134, {3, 5, 9, 11, 21, 25, 35, 45, 49, 59, 61, 65, 67}}},
{14, 150, {3, 5, 9, 11, 21, 25, 35, 43, 53, 57, 67, 69, 73, 75}}}

王守恩

天马行空。来4个数字串。——各位！谈谈您的看法！！

数字串1——2个奇数(≥1)相加(允许重复), 和跑遍偶数(≥2)。a(n)=最大的那个和。
{1, {2, {1}}},
{2, {6, {1, 3}}},
{3, {10, {1, 3, 7}}},
{4, {18, {1, 3, 7, 9}}},
{5, {26, {1, 3, 7, 11, 13}}},
{6, {34, {1, 3, 7, 11, 15, 17}}},
{7, {42, {1, 3, 7, 11, 15, 19, 21}}},
{8, {54, {1, 3, 7, 11, 15, 17, 35, 37}}},
{9, {66, {1, 3, 7, 11, 15, 19, 21, 43, 45}}},
数字串1——2, 6, 10, 18, 26, 34, 42, 54, 66, 82, 94, 110, 130, 146, 162, 186, 210, 234, 258, 282, 306, 330, 362, 394, 426。——没了！

数字串2——2个奇数(≥1)相加(不允许重复), 和跑遍偶数(≥4)。a(n)=最大的那个和。
{2, {4, {1, 3}}},
{3, {8, {1, 3, 5}}},
{4, {14, {1, 3, 5, 9}}},
{5, {20, {1, 3, 5, 9, 15}}},
{6, {28, {1, 3, 5, 7, 13, 21}}},
{7, {36, {1, 3, 5, 7, 13, 21, 29}}},
{8, {46, {1, 3, 5, 7, 9, 17, 27, 37}}},
{9, {56, {1, 3, 5, 9, 15, 21, 25, 27, 29}}},
数字串2——4, 8, 14, 20, 28, 36, 46, 56, 68, 82, 96, 114, 132, 150, 168, 190, 212,——各位！能再来2项？1项也行！！谢谢！！！

数字串3——2个奇数(>1)相加(允许重复), 和跑遍偶数(≥6)。a(n)=最大的那个和。
{1, 6, {3}},
{2, 10, {3, 5}},
{3, 14, {3, 5, 7}},
{4, 22, {3, 5, 9, 11}},
{5, 30, {3, 5, 9, 13, 15}},
{6, 38, {3, 5, 9, 13, 17, 19}},
{7, 46, {3, 5, 7, 13, 19, 21, 23}},
{8, 58, {3, 5, 7, 13, 19, 25, 27, 29}},
{9, 70, {3, 5, 7, 13, 19, 25, 31, 33, 35}},
数字串3——6, 10, 14, 22, 30, 38, 46, 58, 70, 86, 98, 114, 134, 150, 166, 190, 214, 238, 262, 286, 310, 334, 366, 398, 430。——没了！

数字串4——2个奇数(>1)相加(不允许重复), 和跑遍偶数(≥8)。a(n)=最大的那个和。
{2, {8, {3, 5}}},
{3, {12, {3, 5, 7}}},
{4, {18, {3, 5, 7, 11}}},
{5, {24, {3, 5, 7, 11, 17}}},
{6, {32, {3, 5, 7, 11, 17, 23}}},
{7, {40, {3, 5, 7, 11, 17, 23, 31}}},
{8, {50, {3, 5, 7, 11, 17, 23, 31, 39}}},
{9, {60, {3, 5, 7, 11, 17, 23, 31, 39, 49}}},
数字串4——8, 12, 18, 24, 32, 40,50, 60, 72, 86, 100, 118, 136, 154, 172, 194, 216,——各位！能再来2项？1项也行！！谢谢！！！

观察可以有——数字串3 = 数字串1 + 4。数字串4 = 数字串2 + 4。