数字串的通项公式

王守恩

1个编号是1的球,1个球的编号之和=1的整数倍的有1个。
2个编号是1的球,2个编号是2的球,从中取2个球,2个球的编号之和=2的整数倍的有2个。
3个编号是1的球,3个编号是2的球,3个编号是3的球,从中取3个球,3个球的编号之和=3的整数倍的有4个。
4个编号是1的球,4个编号是2的球,4个编号是3的球,4个编号是4的球,从中取4个球,4个球的编号之和=4的整数倍的有10个。
5个编号是1的球,5个编号是2的球,5个编号是3的球,5个编号是4的球,5个编号是5的球,从中取5个球,5个球的编号之和=5的整数倍的有26个。
6个编号是1的球,6个编号是2的球,6个编号是3的球,6个编号是4的球,6个编号是5的球,,6个编号是6的球,从中取6个球,6个球的编号之和=6的整数倍的有多少个?
7个编号是1的球,7个编号是2的球,7个编号是3的球,7个编号是4的球,7个编号是5的球,,7个编号是6的球,7个编号是7的球,从中取7个球,7个球的编号之和=7的整数倍的有多少个?
8个编号是1的球,8个编号是2的球,8个编号是3的球,8个编号是4的球,8个编号是5的球,,8个编号是6的球,8个编号是7的球,8个编号是8的球,从中取8个球,8个球的编号之和=8的整数倍的有多少个?
9个编号是1的球,9个编号是2的球,9个编号是3的球,9个编号是4的球,9个编号是5的球,,9个编号是6的球,9个编号是7的球,9个编号是8的球,9个编号是9的球,从中取9个球,9个球的编号之和=9的整数倍的有多少个?

northwolves

王守恩 发表于 2025-5-3 07:39
1个编号是1的球,1个球的编号之和=1的整数倍的有1个。
2个编号是1的球,2个编号是2的球,从中取2个球,2个球的 ...

f[6]=80

{80,{{1,1,1,1,1,1},{1,1,1,1,2,6},{1,1,1,1,3,5},{1,1,1,1,4,4},{1,1,1,2,2,5},{1,1,1,2,3,4},{1,1,1,3,3,3},{1,1,1,3,6,6},{1,1,1,4,5,6},{1,1,1,5,5,5},{1,1,2,2,2,4},{1,1,2,2,3,3},{1,1,2,2,6,6},{1,1,2,3,5,6},{1,1,2,4,4,6},{1,1,2,4,5,5},{1,1,3,3,4,6},{1,1,3,3,5,5},{1,1,3,4,4,5},{1,1,4,4,4,4},{1,1,4,6,6,6},{1,1,5,5,6,6},{1,2,2,2,2,3},{1,2,2,2,5,6},{1,2,2,3,4,6},{1,2,2,3,5,5},{1,2,2,4,4,5},{1,2,3,3,3,6},{1,2,3,3,4,5},{1,2,3,4,4,4},{1,2,3,6,6,6},{1,2,4,5,6,6},{1,2,5,5,5,6},{1,3,3,3,3,5},{1,3,3,3,4,4},{1,3,3,5,6,6},{1,3,4,4,6,6},{1,3,4,5,5,6},{1,3,5,5,5,5},{1,4,4,4,5,6},{1,4,4,5,5,5},{1,5,6,6,6,6},{2,2,2,2,2,2},{2,2,2,2,4,6},{2,2,2,2,5,5},{2,2,2,3,3,6},{2,2,2,3,4,5},{2,2,2,4,4,4},{2,2,2,6,6,6},{2,2,3,3,3,5},{2,2,3,3,4,4},{2,2,3,5,6,6},{2,2,4,4,6,6},{2,2,4,5,5,6},{2,2,5,5,5,5},{2,3,3,3,3,4},{2,3,3,4,6,6},{2,3,3,5,5,6},{2,3,4,4,5,6},{2,3,4,5,5,5},{2,4,4,4,4,6},{2,4,4,4,5,5},{2,4,6,6,6,6},{2,5,5,6,6,6},{3,3,3,3,3,3},{3,3,3,3,6,6},{3,3,3,4,5,6},{3,3,3,5,5,5},{3,3,4,4,4,6},{3,3,4,4,5,5},{3,3,6,6,6,6},{3,4,4,4,4,5},{3,4,5,6,6,6},{3,5,5,5,6,6},{4,4,4,4,4,4},{4,4,4,6,6,6},{4,4,5,5,6,6},{4,5,5,5,5,6},{5,5,5,5,5,5},{6,6,6,6,6,6}}}

northwolves

1. f[n_] := Sum[EulerPhi[n/k]*Binomial[2 k, k]/(2 n), {k, Divisors@n}]; Table[f[n], {n, 50}]

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{1,2,4,10,26,80,246,810,2704,9252,32066,112720,400024,1432860,5170604,18784170,68635478,252088496,930138522,3446167860,12815663844,47820447028,178987624514,671825133648,2528212128776,9536895064400,36054433810102,136583761444364,518401146543812,1971076362005880,7506908923471954,28634752211620266,109385279303298134,418427080552561516,1602661111666612296,6146007503971075312,23596358977508462296,90692376956955350244,348936088066654523296,1343840109168425292660,5180299766448679532060,19987029597767017004376,77180849825857621779894,298278470246442673742468,1153638014306149018667804,4465167928718123725683748,17294692982395428197325698,67031948061107616221175888,259975635295723029619698462,1008913445455643197454196752}

A003239
Number of rooted planar trees with n non-root nodes: circularly cycling the subtrees at the root gives equivalent trees.

王守恩

不错的一串数——A005920——有这串数——可惜没有这样的条文。

$(\frac{a_{1}+n}{a_{1}})(\frac{a_{2}+n}{a_{2}})(\frac{a_{3}+n}{a_{3}})(\frac{a_{4}+n}{a_{4}})\cdots(\frac{a_{n}+n}{a_{n}})=\frac{2n-1}{n}$

$a_{1},a_{2},a_{3},a_{4},\cdots,a_{n}=正整数.求:a_{1}+a_{2}+a_{3}+a_{4}+\cdots+a_{n}最小值$

$a_{1}+a_{2}+a_{3}+a_{4}+\cdots\cdots+a_{n}最小值=\frac{3 n^3 - 2 n^2 + n}{2}$
譬如。
a(1)=1,\(1=1,(\frac{1+0}{1})=\frac{1+0}{1}\)
a(2)=9,\(9=4+5,(\frac{4+1}{4})(\frac{5+1}{5})=\frac{2+1}{2}\)
a(3)=33,\(33=9+11+13,(\frac{9+2}{9})(\frac{11+2}{11})(\frac{13+2}{13})=\frac{3+2}{3}\)
a(4)=82,\(82=16+19+22+25,(\frac{16+3}{16})(\frac{19+3}{19})(\frac{22+3}{22})(\frac{25+3}{25})=\frac{4+3}{4}\)
a(5)=165,\(165=25+29+33+37+41,(\frac{25+4}{25})(\frac{29+4}{29})(\frac{33+4}{33})(\frac{37+4}{37})(\frac{41+4}{41})=\frac{5+4}{5}\)
a(6)=291,\(291=36+41+46+51+56+61,(\frac{36+5}{36})(\frac{41+5}{41})(\frac{46+5}{46})(\frac{51+5}{51})(\frac{56+5}{56})(\frac{61+5}{61})=\frac{6+5}{6}\)
a(7)=469,\(469=49+55+61+67+73+79+85,(\frac{49+6}{49})(\frac{55+6}{55})(\frac{61+6}{61})(\frac{67+6}{67})(\frac{73+6}{73})(\frac{79+6}{79})(\frac{85+6}{85})=\frac{7+6}{7}\)
a(8)=708,\(708=64+71+78+85+92+99+106+113,(\frac{64+7}{64})(\frac{71+7}{71})(\frac{78+7}{78})(\frac{85+7}{85})(\frac{92+7}{92})(\frac{99+7}{99})(\frac{106+7}{106})(\frac{113+7}{113})=\frac{8+7}{8}\)
a(9)=1017,\(1017=81+89+97+105+113+121+129+137+145,(\frac{81+8}{81})(\frac{89+8}{89})(\frac{97+8}{97})(\frac{105+8}{105})(\frac{113+8}{113})(\frac{121+8}{121})(\frac{129+8}{129})(\frac{137+8}{137})(\frac{145+8}{145})=\frac{9+8}{9}\)

{1, 9, 33, 82, 165, 291, 469, 708, 1017, 1405, 1881, 2454, 3133, 3927, 4845, 5896, 7089, 8433, 9937, 11610, 13461, 15499, 17733, 20172, 22825, 25701, 28809, 32158, 35757}

Table[(3 n^3 - 2 n^2 + n)/2, {n, 29}]

不错的一串数——还可以有更小的吗？

王守恩

northwolves 发表于 2025-5-5 10:30
{1,2,4,10,26,80,246,810,2704,9252,32066,112720,400024,1432860,5170604,18784170,68635478,252088496, ...

Table[FindInstance[{(a + 1)/a + (b + 1)/b - (c + 1)/c == (n + 1)/n, 0 < c < b < a}, {c, b, a}, Integers, 1], {n, 99}]

n可以是这串数吗——A208770——12, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 44, 45, 48, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 75, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 117, 119, 120,

n是这串数——A208770——可有通项公式？谢谢！

n不是这串数——我们可有自己的通项公式？谢谢！！

王守恩

已知：(n + 1)*a[n + 1] = n*a[n] + 2 n + 1,  a[1] = 1, 求：通项公式。

northwolves

王守恩 发表于 2025-5-13 14:56
已知：(n + 1)*a[n + 1] = n*a[n] + 2 n + 1,  a[1] = 1, 求：通项公式。

b(n)=na(n),b(1)=1

b(n+1)=b(n)+2n+1

b(n)=n²

a(n)=n

王守恩

有这样一串数——每次取2个数(允许重复取)相加,  和可以跑遍所有偶数(2, 4, 6, 8, 10, 12, ..........)——还可以“稀”吗？

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, 1461, 1465, 1467, 1477, ..........

A191106—有这串数———通项公式没我们的好——Table[2 FromDigits[IntegerDigits[n, 2], 3] + 1, {n, 0, 64}]

王守恩

northwolves 发表于 2025-5-13 15:55
b(n)=na(n),b(1)=1

b(n+1)=b(n)+2n+1

用 2，3，+，×，( )。譬如：
a(1)=0,
a(2)=1,{2},
a(3)=1,{3},
a(4)=2,{2*2},
a(5)=2,{2+3},
a(6)=2,{2*3},
a(7)=3,{2*2+3},
a(8)=3,{2*2*2},
a(9)=2,{3*3},
a(10)=3,{2(2+3)},
a(11)=3,{3*3+2},
a(12)=3,{3*3+3},
a(13)=4,{3*3+2*2}
a(14)=4,{3*3+2+3},
a(15)=3,{3(2+3},
a(16)=4,{2*2*2*2},
a(17)=4,{3(2+3)+2},
a(18)=3,{2*3*3},
a(19)=5,{2*2*2*2+3},
a(20)=4,{2*3*3+2},
a(21)=4,{2*3*3+3},
a(22)=4,{2(3*3+2)},
a(23)=5,{2*3*3+2+3}
a(24)=4,{2*3(2+2)},
a(25)=4,{(2+3)(2+3)},
a(26)=5,{2*3(2+2)+2},
a(27)=3,{3*3*3},
a(28)=5,{2(3*3+2+3)},
a(29)=4,{3*3*3+2},
a(30)=4,{3*3*3+3)},
a(31)=5,{3*3*3+2*2},
a(32)=5,{3*3*3+2+3},
a(33)=4,{3(3*3+2)}
a(34)=5,{2((3(2+3)+2)},
a(35)=5,{3(3*3+2)+2},
a(36)=4,{3(3*3+3)},
a(37)=6,{3(3*3+2)+2*2},
a(38)=5,{3(3*3+3)+2},


得到一串数：0, 1, 1, 2, 2, 2, 3, 3, 2, 3, 3, 3, 4, 4, 3, 4, 4, 3, 5, 4, 4, 4, 5, 4, 4, 5, 3, 5, 4, 4, 5, 5, 4, 5, 5, 4, 6, 5, ——OEIS没有这串数？——可有通项公式？谢谢！

王守恩

第1个问题。
2^1-1!=1,
1*2^2-2!=2,
2*2^3-3!=10,
10*2^4-4!=136,
136*2^5-5!=4232,
4232*2^6-6!=270128,
270128*2^7-7!=34571344,
34571344*2^8-8!=8850223744,
......
得到一串数——{1, 2, 10, 136, 4232, 270128, 34571344, 8850223744, 4531314194048, 4640065731076352, 9502854617204452096, 38923692512068956783616, 318862889058868887744361472,
5224249574340507856716440066048, 171188210051989761448883000409892864, 11218990533967201006313996293939948847104, 1470495527268148970299588122238941287859519488}——求通项公式——问题1。

第2个问题。
2^1-1=1,
1*2^2-3=1,
1*2^3-6=2,
2*2^4-10=22,
22*2^5-15=689,
689*2^6-21=44075,
44075*2^7-28=5641572,
5641572*2^8-36=1444242396,
1444242396*2^9-45=739452106707,
.......
得到一串数——{1, 1, 2, 22, 689, 44075, 5641572, 1444242396, 739452106707, 757198957267913, 1550743464484685758, 6351845230529272864690, 52034316128495803307540389,
852530235449275241390741733271, 27935710755201851109891825115824008, 1830794740052908514337870650790642188152, 239965928168214824791293381940431052885458791}——求通项公式——问题2。

第3个问题。
136/22=6.18182,
4232/689=6.14224,
270128/44075=6.12883,
34571344/5641572=6.12796,
......
5224249574340507856716440066048/852530235449275241390741733271=6.12793465511211873361212958209,
......
262345168553487441312403456944635391129575147770993115164671264739968851694392029493592064/
42811352163266089318903999189948042642370639369565874377218194446248920891354503995981524=6.12793465511211873361208276053,
......
这个6.12793465511211873361208276053, 最后=？？？——问题3。