do  -- find some Chowla numbers ( Chowla n = sum of divisors of n exclusing n and 1 )

    -- returns the divisor sums of [1 n], the sums exclude 1 and n
    local function sumDivisors( n : number ) : table
        local sums <const> = {}
        for i = 1, n do sums[ i ] = 0 end
        for i = 2, n do
            for j = i + i, n, i do sums[ j ] += i end
        end
        return sums
    end

    local ds <const> = sumDivisors( 10_000_000 )

    local function chowla( n : number ) : number   -- returns the Chowla number of n
        local sum = 0
        if n > 0 then
            if n <= # ds then
                sum = ds[ n ]
            else
                for i = 2, math.floor( math.sqrt( n ) ) do
                    if n % i == 0 then
                        local j <const> = n // i
                        sum += i + if i == j then 0 else j end
                    end
                end
            end
        end
        return sum
    end

    -- first 37 chowla numbers
    for n = 1, 37 do print( $"chowla({n}) = {chowla(n)}" ) end

    -- number of primes up to 10_000_000
    local count, power = 0, 100
    for n = 2, 10_000_000 do
        if chowla( n ) == 0 then count += 1 end
        if n % power == 0 then
            print( string.format( "There are %6d primes < %d", count, power ) )
            power *= 10
        end
    end

    -- perfect numbers to 35_000_000
    local perfectMax <const> = 35_000_000
    count = 0
    local k, kk, p = 2, 3, 6       -- p is k * kk
    while p <= perfectMax do
        if chowla( p ) == p - 1 then
            print( string.format( "        %8d is a perfect number", p ) )
            count += 1
        end
        k = kk + 1; kk += k; p = k * kk
    end
    print( $"There are {count} perfect numbers < {perfectMax}" )
end