def moebius_sum(n):
r"""Returns the value of the sum of moebius(k) for k = 1, 2, ..., n.
Parameters
----------
n : int (n > 0)
Returns
-------
s : int
Raises
------
ValueError
If n <= 0.
Notes
-----
The formula used here is based on the relationship \sum_{d = 1}^{n}
M(n/d) = 1, where M(n) = \sum_{k = 1}^{n} moebius(k). We convert this
to a recursive formula, whereby using some form of Dirichlet's
hyperbola method and memoizing the recursion gives us a sublinear
runtime.
Examples
--------
>>> moebius_sum(10)
-1
>>> sum(lists.moebius_list(10))
-1
>>> moebius_sum(0)
Traceback (most recent call last):
...
ValueError: moebius_sum: Must have n > 0.
"""
if n <= 0:
raise ValueError("moebius_sum: Must have n > 0.")
sqrt = int(n**(0.5))
mu_list = lists.moebius_list(sqrt)
# for i <= sqrt(n), compute the sum directly
cache = {1: 1}
for i in xrange(2, sqrt + 1):
cache[i] = cache[i - 1] + mu_list[i]
def M(n):
if n in cache:
return cache[n]
sqrt_n = int(n**(0.5)) + 1
value = n - 1
for d in xrange(2, sqrt_n):
value += M(n//d) - M(d - 1)
value += mu_list[d]*(n//d - d)
cache[n] = 1 - value
return cache[n]
return M(n)
def test_moebius_list(self):
self.assertRaisesRegexp(ValueError, "moebius_list: Must have n > 0.", moebius_list, 0)
values = [
0, 1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, -1, -1,
0, 1, 1, 1, 0, -1, 1, 1, 0, -1, -1, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, 1, 0, 0,
1, -1, -1, 0, 1, -1, -1, 0, -1, 1, 0, 0, 1, -1, -1, 0, 0, 1, -1, 0, 1, 1, 1, 0, -1, 0, 1, 0, 1, 1, 1, 0, -1,
0, 0, 0, -1, -1, -1, 0, -1, 1, -1, 0, -1, -1, 1, 0, -1, -1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, -1, 0, 1,
-1, -1, 0, 1, 1, 0, 0, -1, -1, -1, 0, 1, 1, 1, 0, 1, 1, 0, 0, -1, 0, -1, 0, 0, -1, 1, 0, -1, 1, 1, 0, 1, 0,
-1, 0, -1, 1, -1, 0, 0, -1, 0, 0, -1, -1, 0, 0, 1, 1, -1, 0, -1, -1, 1, 0, 1, -1, 1, 0, 0, -1, -1, 0, -1, 1,
-1, 0, -1, 0, -1, 0
]
self.assertEqual(moebius_list(200), values)
