考拉茲級數函式

考拉茲級數函式是考拉茲猜想的拓展級數函式

定義

它是個疊代循環級數,但是不是所有的考拉茲級數是收斂的

套用

它是考拉茲猜想1927年提出的猜想,一個整數n,如果是偶數則除以2,如果是奇數則乘以3再加1,如此循環計算結果是4,2,1的循環,研究它對於一些自然現象是有幫助的。下面是考拉茲級數的通項表達式

考拉茲級數函式 考拉茲級數函式

這是原函式,k級通項式為

考拉茲級數函式 考拉茲級數函式

下面是vba程式代碼,F(n,a)函式是求級數值,k(i,a)是層級數

Function F(N As Variant,a as Variant ) As Variant

Dim b As Variant

Dim c As Variant

b = (-1) ^ (N + 1)

c =( 1-(-1) ^ N)/2

F = c + N * (1+(a-1) ^ c) ^ b

End Function

Sub 求考拉茲級數函式()

Dim k As Variant

Dim i As Integer

Dim j As Integer

Dim m As Variant

k = InputBox(Prompt:="輸入任意整數")

m = F(k,3): j = 1

Do While m > 1

m = F(m): j = j + 1

Loop

m = 1

MsgBox "總計考拉茲" & j & "級"

End

End Sub

以下函式是用於excel中求考拉茲級數的

下面兩個函式必須放在一個模組之中

Function K(I As Variant,a as Variant ) As Integer

Dim j As Integer

Dim m As Variant

m = F(I,a): j = 1

Do While m > 1

m = F(m): j = j + 1

Loop

m = 1

K = j

End Function

Function F(N As Variant,a as Variant ) As Variant

b = (-1) ^ (N + 1)

c =( 1-(-1) ^ N)/2

F = c + N * (1+(a-1) ^ c) ^ b

End Function

EXCEL中

使用時直接輸入 =k(m,a)即可求出考拉茲級數=F(m,a)即可求出考拉茲級數

當a不等於3時,函式多數是發散的。基本很少有循環的規律性.

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