In Python:

def flavius(n, p):
  if n == 1: return 0
  if n < p: return (flavius(n - 1, p) + p) % n
  q, r = n/p, n%p
  i = flavius(n - q, p) - r
  if i < 0: return i + n
  return i + i / (p - 1)

Or as a piecewise function:

f(n, p) = { 0,                      n = 1
          ( f(n-1, p) + p (mod n),  n < p
          ( f(n-q, p) - r + n,      f(n-q, p) < r
          ( f(n-q, p) x (p/)   otherwise

where q is the quotient (n-r)/p and r is the remainder n (mod p).

Note that this function returns the zero-based index of the survivor.