欧拉函数迭代问题

wsc810

以下对卡迈克尔数561做重新计算

$\varphi(561)=\varphi(3*11*17)=\varphi(3)\varphi(11)\varphi(17)=2*2^3*2^4=2^8$

$\varphi(560)=\varphi(2^4*5*7)=\varphi(2^4)\varphi(5)\varphi(7)=2^4*2^2*2^2=2^8$


卡迈克尔数1105

$\varphi(1105)=\varphi(5*13*17)=\varphi(5)\varphi(13)\varphi(17)=2^2*2^3*2^4=2^9$

$\varphi(1104)=\varphi(2^4*3*23)=\varphi(2^4)\varphi(3)\varphi(23)=2^4*2*2^4=2^9$


卡迈克尔数1729

$\varphi(1729)=\varphi(7*13*19)=\varphi(7)\varphi(13)\varphi(19)=2^2*2^3*2^3=2^8$

$\varphi(1728)=\varphi(2^6*3^3)=\varphi(2^6)\varphi(3^3)=2^6*2^3=2^9$

wsc810

重算 2047


$\varphi(2047)=\varphi(23)\varphi(89)=\varphi(22)*\varphi(88)=\varphi(16)\varphi(11^2)=2^4*(2^3)^2=2^10$

$\varphi(2046)=\varphi(2)\varphi(1023)=\varphi(2)\varphi(3)\varphi(11)\varphi(31)=2*2*2^3*2^4=2^9$


4181

$\varphi(4181)=\varphi(37)\varphi(113)=\varphi(36)\varphi(112)=\varphi(36)\varphi(16)\varphi(7)=2^4*2^4*2^2=2^10$

$\varphi(4180)=\varphi(4)\varphi(5)\varphi(11)\varphi(19)=2^2*2^2*2^3*2^3=2^10$

wsc810

编写一个Mathematica函数算吧，手工计算太麻烦

hujunhua

第1种定义的Mathematica程序

1. f[n0_]:=
   
2. Module[{n=n0}, f[1]=1; f[2]=2;
   
3.        inner[{x_List, y_List}]:=Inner[Power, f/@x, y, Times];
   
4.        f[n_]:=Which[PrimeQ@n, f[n-1],
   
5.                       OddQ@n, inner@Transpose@FactorInteger@n,
   
6.                      EvenQ@n, 2 inner@Transpose@FactorInteger@NestWhile[#/2&, n, EvenQ]];
   
7.        f[n]]

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Table[{n, f[n]}, {n, 25}]//Transpose//MatrixForm输出

第2种定义的Mathematica程序

1. ff[n0_]:=
   
2. Module[{n=n0}, ff[1]=1; ff[2]=2;
   
3.        inner[{x_List, y_List}]:=Inner[Power, ff/@x, y, Times];
   
4.        ff[n_]:=If[PrimeQ@n, ff[n-1], inner@Transpose@FactorInteger@n];
   
5.        ff[n]]

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Table[{n, ff[n]}, {n, 25}]//Transpose//MatrixForm输出


缩进控制不准，懒得调了，将就看吧。

zeroieme

1. 100//Range//{#,#//Which[#==1,0,#==2,1,PrimeQ[#],#0[#-1],True,{#0,#//FactorInteger}//Function[{RecursionFunction,nSet},RecursionFunction[#[[1]]]#[[2]]&/@nSet//Total]@@#&]&}&/@#&
   
2. %//ListPlot

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