Sorry, but M_1 is simply the sum of divisors, and I don't think that was ever a mystery. Specializing the notation from the paper for M_a, to a=1, and writing pythonic with finite bounds for clarity...

  M_1(n) = sum(
    m
    for m in range(1, n+1)
    for s in range(1, n+1)
    if m*s = n
  )