(オイラーの)totient関数の性質

totient.jl

# https://gist.github.com/yamasushi/873992c10d8ac74750f3fa9d3ec039f4
# https://qiita.com/yamasushi/items/fc0d4c77d26bb5d05ed3

using DataFrames
using Primes

n=30
@show(n)
gcds = gcd.(n,1:n)
df=DataFrame(
    i          = 1:n
    , gcd_n_i  = gcds
    , div_n_gcd= div.(n  ,gcds)
    , div_i_gcd= div.(1:n,gcds)
    )
sort!(df,:div_n_gcd)

gcd_n_i   =unique(df.gcd_n_i)
div_n_gcd =unique(df.div_n_gcd)

z_zms =
    map(div_n_gcd) do m
        m , 
        rename(df[ df.div_n_gcd .== m , [:i,:div_i_gcd]] , 
            :div_i_gcd=>Symbol("(Z/Z$m)×"))
    end

totient_gcd_n_i  =totient.(gcd_n_i)
totient_div_n_gcd=totient.(div_n_gcd)
println("(n,i)=$gcd_n_i , totients=$totient_gcd_n_i , sum=$(sum(totient_gcd_n_i))")
println("n/(n,i)=$div_n_gcd , totients=$totient_div_n_gcd , sum=$(sum(totient_div_n_gcd))")

rename!(df,
    :gcd_n_i  =>Symbol("(n,i)")   ,
    :div_n_gcd=>Symbol("n/(n,i)") ,
    :div_i_gcd=>Symbol("i/(n,i)"))

println("")
show(df)
println("\n")

for (m,z_zm) in z_zms
    #@show(z_zm)
    println("m=$m , totient=$(totient(m))")
    show(z_zm)
    println("\n")
end