Optimizing implementation of sigma function

sigma.jl

using Primes
using BenchmarkTools

function σ{T}(k, n, ::Type{T} = Int)
    result = ones(T, n - 1)
    
    for p in primes(n)
        bound = round(Int, log(n) / log(p), RoundDown)
        pᵏ = convert(T, p)^k
        pᵏⁱ = one(T)
        si = pᵏⁱ
        pi = 1
        
        for i in 0:bound
            for pj in eachindex(result)
                if pi * pj < n
                    @inbounds result[pi*pj] = si * result[pj]
                else
                    break
                end
            end
            
            pi *= p
            pᵏⁱ *= pᵏ
            si += pᵏⁱ
        end
    end

    return result
end


# Original code from https://stackoverflow.com/q/49021937/1346276

thefunc(i) = (a::Int, b::Int)-> sum(BigInt(a)^(k*i) for k in 0:b)

function powsfunc(x, n, func)
    bound = convert(Int, floor(log(n)/log(x)))
    return Dict{Int, Number}(x^i => func(x, i) for i in 0:bound)
end

dictprod2(d1, d2, bd)=Dict(i*j=> d1[i]*d2[j] for i in keys(d1), j in keys(d2) if i*j<bd)

function dosigma(k, n)
    primefunc = thefunc(k)
    combfunc = (a, b) -> dictprod2(a, b, n)
    allprimes = [powsfunc(p, n, primefunc) for p in primes(n)]
    trivdict = Dict{Int, Number}(1=>1)
    theresult = reduce(combfunc, trivdict, allprimes)
    return theresult
end


function test(;k = 0, n = 100, T = Int)
    a = @btime σ($k, $n, $T)
    b = dosigma(k, n)
    @assert all(a[i] == b[i] for i = 1:(n-1))
end