求数列的通项公式

ysr

求如下数列的通项公式：11，17，41，101，221，431，767，……

中国上海市

11，17，41，101，221，431，767
6，24，60，120，210，336
18，36，60，90，126
18，24，30，36
6，6，6
后一项依次减前一项，到第四遍的时候就是个常数了，也就是说到第三遍的时候就是个等差数列了，很好求的

luyuanhong

ysr

太感谢了，非常好！谢谢陆教授，谢谢中国上海市先生！陆教授的太好了，原来公式就是：an=11+(n-1)n(n+1)(n+2)/4,n>=2.

ysr

谢谢教授！我用公式编了程序，可以不用从头递推直接求中间任意1个

ysr

这是我推测的数列，原来没有通项公式，谢谢教授！猜测此数列有这样的特性：其中的素数及大因子合数p输入公式p+n（n+1）前p-1项内素数密度较高，主楼的可以叫元帅数列，公式p+n（n+1）可以叫欧拉数列

ysr

如下125为元帅数列程序，输入100则得到前100项（素数合数是分开的），括号中数值为项数：

如下123为欧拉数列程序，输入素数则得到前50项，（素数合数是分开的），括号中数值为项数：

ysr

1与100之间的以数列11+(n-1)n(n+1)(n+2)/4,（n>=2.）的素数开头最多有40个连续素数:
3
/17/19/23/29/37/47/59/73/89/107/127/149/173/199/227/257
/41/43/47/53/61/71/83/97/113/131/151/173/197/223/251/281/313/347/383/421
/461/503/547/593/641/691/743/797/853/911/971/1033/1097/1163/1231/1301/1373/1447/1523/1601

程序代码如下：（仅发主程序）
Private Sub Command1_Click()
Dim a, b
a = Val(Text1)
a1 = a
Q = Val(Text2)
m = Sqr(Q)
If Right(a, 1) Mod 2 = 0 Then
a = a + 1
Else
a = a
End If

Do While InStr(fenjieyinzi0(Val(a)), "*") <> 0
a = a + 2
Loop
s101 = 1
Do While a <= Q
b = a + s2
d = fenjieyinzi0(Val(b))
s = 0
s3 = ""
Do While s < 40 And InStr(d, "*") = 0
s4 = s + 1
If s4 Mod 20 = 0 Then
s3 = s3 & "/" & b & vbCrLf
Else
s3 = s3 & "/" & b
End If
s = s + 1
s2 = s * (s + 1)
b = a + s2
d = fenjieyinzi0(Val(b))
Loop
If s > 2 Then
Text3 = Text3 & s3 & vbCrLf
ElseIf InStr(fenjieyinzi0(Val(a)), "*") = 0 Then
Text3 = Text3 & a & vbCrLf
End If
If s > 1 Then
s5 = s5 & "/" & s
Else
s5 = s5
End If
Print s5
s101 = s101 + 1
a = 11 + (s101 - 1) * s101 * (s101 + 1) * (s101 + 2) / 4
s2 = 0
Loop
s6 = Mid(s5, 2)
     s7 = max(Trim(s6))
Combo1 = a1 & "与" & Q & "之间的数列11+(n-1)n(n+1)(n+2)/4,（n>=2.）的素数开头最多有" & s7 & "个连续素数:" & vbCrLf & Text3

     
Text4 = s5
End Sub

Private Sub Command2_Click()
Text1 = ""
Text2 = ""
Text3 = ""
Combo1 = ""
Form1.Cls
End Sub
Private Function max(ByVal lists As String) As String
Dim temp As Long
Dim a() As String
a = Split(lists, "/")
Dim b As Long

For b = 0 To UBound(a)
If temp < a(b) Then temp = a(b)
Next
max = temp
End Function

ysr

18371与10000000000之间的数列11+(n-1)n(n+1)(n+2)/4,（n>=2.）的素数开头最多有3个连续素数:
18371
122861
188801
215771
548351
671591
1624361
1897517
2925821
3347081
4062251
5501381
6530591
17518421
/19101281/19101283
/20789051/20789053/20789057
49287431
70299851
86778551
112031651
131687111
201512231
226517561
259516001
265347821
283434071
/428717741/428717743
/437207201/437207203
/529207031/529207033
549222671
559440767
/669645017/669645019
755947541
782069201
808862051
953296511
984359261
2177575571
2384124767
/2508557321/2508557323
2670874091
2840943311
3282201401
/3687586361/3687586363
/4083529517/4083529519
4510532771
4710694601
5753905181
6431959811
7029900191
/8747393267/8747393269

所以，以该数列中的素数打头的多项式，并不能输出太多的连续素数。

ysr

96637359101与10000000000000之间的数列11+(n-1)n(n+1)(n+2)/4,（n>=2.）的素数开头最多有4个连续素数:
96637359101
259635597491
271138383401
277571449361
278654655017
280830574301
281923300271
288546774401
/289661931017/289661931019
293026801091
303297474911
306780285017
317408855501
344551977221
373399823171
385708691981
402590884511
448769359517
470877988241
474101791061
475719885911
487162818881
508960248821
527968631621
529722680231
/569429460641/569429460643
580635618041
605482406267
611327733761
657559620911
/676345872017/676345872019/676345872023/676345872029
744148632251
771717447161
774048919811
824241555017
902747967041
/910632187181/910632187183
931909310681
964521392111
995160868517
1020767719241
1052732430767
1162613358281
1168959923051
1191377707517
1240500319517
/1280879565851/1280879565853
1343286795017
/1346819436161/1346819436163
1393380752651
1397011712267
1437422354561
1478703424391
1520867332001
/1544242398311/1544242398313
1563926575481
1636351360811
1694398157801
1698602619731
1779984239771
1788717667481
1815110854871
/1886921190767/1886921190769/1886921190773
1970231918861
1974939760961
2046574872821
2299500804521
2374340451221
2385179082017
/2417917706201/2417917706203
2439929910767
2686171853081
2727955980767
2764157293211
2812984745231
2831460780341
3099802151261
3234082504391
3316716625421
3330641425031
3415113240017
3457946657591
3530229511001
3552133668821
3715894279901
3784493617517
/3948250467431/3948250467433
4216248275861
4359516410981
4419558073361
4550305457891
4621212239711
4738199204351
4931852704517
4941219946121
5026126819517
/5257897153121/5257897153123
5336906220461
5426852123171
5436915826691
5672268630767
5682671691071
5893753994741
6012326068031
6132678304211
6243649114181
6390154543511
6493081163267
6620563145081
/6773633467031/6773633467033
6809331071441
6881149384841
7149353499767
7476344538401
7920720712517
8204904800831
8301310558811
8384618080031
8496668244131
8781706962641
8854073360831
9464479997171
9479751445331
9679966788971
9962403287267