def multiplicity(p, n):
"""
Find the greatest integer m such that p**m divides n.
Examples
========
>>> from sympy.ntheory import multiplicity
>>> [multiplicity(5, n) for n in [8, 5, 25, 125, 250]]
[0, 1, 2, 3, 3]
"""
from residue_ntheory import int_tested
p, n = int_tested(p, n)
if p == 2:
return trailing(n)
if p < 2:
raise ValueError('p must be an integer, 2 or larger, but got %s' % p)
if p == n:
return 1
m = 0
n, rem = divmod(n, p)
while not rem:
m += 1
if m > 5:
# The multiplicity could be very large. Better
# to increment in powers of two
e = 2
while 1:
ppow = p**e
if ppow < n:
nnew, rem = divmod(n, ppow)
if not rem:
m += e
e *= 2
n = nnew
continue
return m + multiplicity(p, n)
n, rem = divmod(n, p)
return m
def multiplicity(p, n):
"""
Find the greatest integer m such that p**m divides n.
Examples
========
>>> from sympy.ntheory import multiplicity
>>> [multiplicity(5, n) for n in [8, 5, 25, 125, 250]]
[0, 1, 2, 3, 3]
"""
from residue_ntheory import int_tested
p, n = int_tested(p, n)
if p == 2:
return trailing(n)
if p < 2:
raise ValueError('p must be an integer, 2 or larger, but got %s' % p)
if p == n:
return 1
m = 0
n, rem = divmod(n, p)
while not rem:
m += 1
if m > 5:
# The multiplicity could be very large. Better
# to increment in powers of two
e = 2
while 1:
ppow = p**e
if ppow < n:
nnew, rem = divmod(n, ppow)
if not rem:
m += e
e *= 2
n = nnew
continue
return m + multiplicity(p, n)
n, rem = divmod(n, p)
return m
def totient(n):
"""
Calculate the Euler totient function phi(n)
>>> from sympy.ntheory import totient
>>> totient(1)
1
>>> totient(25)
20
See Also
========
divisor_count
"""
from residue_ntheory import int_tested
n = int_tested(n)
if n < 1:
raise ValueError("n must be a positive integer")
factors = factorint(n)
t = 1
for p, k in factors.iteritems():
t *= (p - 1) * p**(k - 1)
return t
def totient(n):
"""
Calculate the Euler totient function phi(n)
>>> from sympy.ntheory import totient
>>> totient(1)
1
>>> totient(25)
20
See Also
========
divisor_count
"""
from residue_ntheory import int_tested
n = int_tested(n)
if n < 1:
raise ValueError("n must be a positive integer")
factors = factorint(n)
t = 1
for p, k in factors.iteritems():
t *= (p-1) * p**(k-1)
return t
