def compute():
LIMIT = 10**7
smallestprimefactor = eulerlib.list_smallest_prime_factors(LIMIT)
ans = 0
for i in range(1, LIMIT + 1):
# Compute factorization as coprime prime powers. e.g. 360 = {2^3, 3^2, 5^1}
factorization = []
j = i
while j != 1:
p = smallestprimefactor[j]
q = 1
while True:
j //= p
q *= p
if j % p != 0:
break
factorization.append(q)
solns = [0]
modulus = 1
for q in factorization:
# Use Chinese remainder theorem; cache parts of it
recip = eulerlib.reciprocal_mod(q % modulus, modulus)
newmod = q * modulus
solns = [((0 + (x ) * recip * q) % newmod) for x in solns] + \
[((1 + (x - 1) * recip * q) % newmod) for x in solns]
modulus = newmod
ans += max(solns)
return str(ans)
def compute():
LIMIT = 10**7
smallestprimefactor = eulerlib.list_smallest_prime_factors(LIMIT)
ans = 0
for i in range(1, LIMIT + 1):
# Compute factorization as coprime prime powers. e.g. 360 = {2^3, 3^2, 5^1}
factorization = []
j = i
while j != 1:
p = smallestprimefactor[j]
q = 1
while True:
j //= p
q *= p
if j % p != 0:
break
factorization.append(q)
solns = [0]
modulus = 1
for q in factorization:
# Use Chinese remainder theorem; cache parts of it
recip = eulerlib.reciprocal_mod(q % modulus, modulus)
newmod = q * modulus
solns = [((0 + (x ) * recip * q) % newmod) for x in solns] + \
[((1 + (x - 1) * recip * q) % newmod) for x in solns]
modulus = newmod
ans += max(solns)
return str(ans)
def compute():
LIMIT = 10**7
smallestprimefactor = eulerlib.list_smallest_prime_factors(LIMIT)
# Maximum size of set of prime factors where the product of the set <= LIMIT.
# This is important because the number of solutions for n is 2^N,
# where N is the number of distinct prime factors of n.
maxnumprimefactors = 0
prod = 1
for i in range(2, len(smallestprimefactor)):
if smallestprimefactor[i] == i: # i is prime
if LIMIT // prod < i:
break
prod *= i
maxnumprimefactors += 1
ans = 0
# Temporary arrays
solns = [0] * (2**maxnumprimefactors)
newsolns = [0] * (2**maxnumprimefactors)
for i in range(1, LIMIT + 1):
# Compute factorization as coprime prime powers. e.g. 360 = {2^3, 3^2, 5^1}
factorization = []
j = i
while j != 1:
p = smallestprimefactor[j]
q = 1
while True:
j //= p
q *= p
if j % p != 0:
break
factorization.append(q)
solns[0] = 0
solnslen = 1
modulus = 1
for q in factorization:
# Use Chinese remainder theorem; cache parts of it
recip = eulerlib.reciprocal_mod(q % modulus, modulus)
newmod = q * modulus
newsolnslen = 0
for j in range(solnslen):
newsolns[newsolnslen] = (0 + (
(solns[j] - 0 + modulus) * recip % modulus) * q) % newmod
newsolnslen += 1
newsolns[newsolnslen] = (1 + (
(solns[j] - 1 + modulus) * recip % modulus) * q) % newmod
newsolnslen += 1
solnslen = newsolnslen
modulus = newmod
# Flip buffers
solns, newsolns = newsolns, solns
ans += max(solns[:solnslen])
return str(ans)
def compute():
ans = 0
primes = eulerlib.list_primes(2000000)
for i in itertools.count(2):
p = primes[i]
q = primes[i + 1]
if p > 1000000:
break
k = 1
while k < p:
k *= 10
m = (q - p) * eulerlib.reciprocal_mod(k % q, q) % q
ans += m * k + p
return str(ans)
def compute(): ans = 0 primes = eulerlib.list_primes(2000000) for i in itertools.count(2): p = primes[i] q = primes[i + 1] if p > 1000000: break k = 1 while k < p: k *= 10 m = (q - p) * eulerlib.reciprocal_mod(k % q, q) % q ans += m * k + p return str(ans)
def chinese_remainder_theorem(a, p, b, q):
return (a + (b - a) * eulerlib.reciprocal_mod(p % q, q) * p) % (p * q)
def chinese_remainder_theorem(a, p, b, q): return (a + (b - a) * eulerlib.reciprocal_mod(p % q, q) * p) % (p * q)
def s(p):
return (p - 3) * eulerlib.reciprocal_mod(8 % p, p) % p
def s(p): return (p - 3) * eulerlib.reciprocal_mod(8 % p, p) % p