def euler_phi_weighted_sum(n):
"""Returns the value of the sum of k*euler_phi(k) for k = 1, 2, ..., n.
Parameters
* n: int (n > 0)
Returns:
* s: int
Raises:
* ValueError: If n <= 0.
Examples:
>>> euler_phi_weighted_sum(100)
203085
>>> sum(k*euler_phi(k) for k in xrange(1, 101))
203085
>>> euler_phi_weighted_sum(0)
Traceback (most recent call last):
...
ValueError: euler_phi_weighted_sum: Must have n > 0.
Details:
Let T(n) = \sum_{k = 1}^{n} k*euler_phi(k). We use the relationship
\sum_{d = 1}^{n} d*T(n/d) = n*(n + 1)*(2*n + 1) / 6 to form a recurrence
for T(n). Using a form of Dirichlet's hyperbola method and memoizing the
recursion gives us a sublinear runtime.
"""
if n <= 0:
raise ValueError("euler_phi_weighted_sum: Must have n > 0.")
sqrt = int(n**(0.5))
totients = lists.euler_phi_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] + i*totients[i]
def T(n):
if n in cache:
return cache[n]
sqrt_n = int(n**(0.5)) + 1
s1 = sum(d*totients[d]*(n//d - d + 1)*(n//d + d)//2 for d in xrange(1, sqrt_n))
s2 = sum(d*(T(n//d) - T(d - 1)) for d in xrange(2, sqrt_n))
s3 = sum(d*d*totients[d] for d in xrange(1, sqrt_n))
cache[n] = n*(n + 1)*(2*n + 1)//6 - (s1 + s2 - s3)
return cache[n]
return T(n)
def test_euler_phi_list(self):
self.assertRaisesRegexp(ValueError, "euler_phi_list: Must have n > 0.", euler_phi_list, 0)
values = [
0, 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30,
16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, 32, 24, 52, 18, 40, 24, 36, 28,
58, 16, 60, 30, 36, 32, 48, 20, 66, 32, 44, 24, 70, 24, 72, 36, 40, 36, 60, 24, 78, 32, 54, 40, 82, 24, 64,
42, 56, 40, 88, 24, 72, 44, 60, 46, 72, 32, 96, 42, 60, 40, 100, 32, 102, 48, 48, 52, 106, 36, 108, 40, 72,
48, 112, 36, 88, 56, 72, 58, 96, 32, 110, 60, 80, 60, 100, 36, 126, 64, 84, 48, 130, 40, 108, 66, 72, 64,
136, 44, 138, 48, 92, 70, 120, 48, 112, 72, 84, 72, 148, 40, 150, 72, 96, 60, 120, 48, 156, 78, 104, 64,
132, 54, 162, 80, 80, 82, 166, 48, 156, 64, 108, 84, 172, 56, 120, 80, 116, 88, 178, 48, 180, 72, 120, 88,
144, 60, 160, 92, 108, 72, 190, 64, 192, 96, 96, 84, 196, 60, 198, 80
]
self.assertEqual(euler_phi_list(200), values)
def euler_phi_sum(n):
r"""Returns the value of the sum of euler_phi(k) for k = 1, 2, ..., n.
Parameters
----------
n : int (n > 0)
Returns
-------
s : int
Raises
------
ValueError
If n <= 0.
Notes
-----
Let S(n) = \sum_{k = 1}^{n} \phi(k). We use the identity S(n) = n*(n +
1)/2 - \sum_{d = 2}^{n} S(n/d) to compute the value. By applying a
variant of Dirichlet's Hyperbola Method, we are able to cut the limits
of summation, and by memoizing the recursion, we are able to achieve a
sublinear runtime. See Section 4.2 of "The Prime Numbers and Their
Distribution" by Mendes, et al for the Hyperbola Method. See section
4.9 equation 4.60 in "Concrete Mathematics" by Graham, et al for a
proof of the identity for S(n).
Examples
--------
>>> euler_phi_sum(10)
32
>>> sum(lists.euler_phi_list(10))
32
>>> euler_phi_sum(10**5)
3039650754
>>> euler_phi_sum(0)
Traceback (most recent call last):
...
ValueError: euler_phi_sum: Must have n > 0.
"""
if n <= 0:
raise ValueError("euler_phi_sum: Must have n > 0.")
sqrt = int(n**(0.5))
phi_list = lists.euler_phi_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] + phi_list[i]
def S(n):
if n in cache:
return cache[n]
sqrt_n = int(n**(0.5))
value = n - sqrt_n*cache[sqrt_n]
for d in xrange(2, sqrt_n + 1):
value += (S(n//d) + phi_list[d]*(n//d))
cache[n] = n*(n + 1)//2 - value
return cache[n]
return S(n)
