def solve():
longestLength = 0
smallestMember = None
seen = set()
sieve(10**6)
for n in xrange(2, 1000001):
if n in seen:
continue
chainIndices = {n: 0}
i = 0
while n > 1:
n = aliquot(n)
i += 1
if n > 1000000: break
# This ^^ technically doesn't comply with a _prima facie_
# interpretation of the problem, but it makes the program terminate
# in under a minute with the correct answer, so...
if n in chainIndices:
i2 = chainIndices[n]
if i - i2 > longestLength:
chain = [m for m,j in chainIndices.iteritems() if j >= i2]
if all(m <= 1000000 for m in chain):
longestLength = i - i2
smallestMember = min(chain)
break
elif n in seen:
break
seen.add(n)
chainIndices[n] = i
return smallestMember
def solve():
sieve(500)
P = [Fraction(2 if isPrime(i) else 1, 3) for i in xrange(1, 501)]
N = [1 - p for p in P]
state = [Fraction(1, 500)] * 500
for primal in [P if c == 'P' else N for c in 'PPPPNNPPPNPPNPN']:
state = jump(map(operator.mul, state, primal))
return sum(state)
def solve():
sieve(12000)
cache = [0,0,0,0] + [None] * 12001
qty = 0
for d in xrange(4, 12001):
cache[d] = twixtQty(d) - sum(cache[div] for div in divisors(d)
if div != d)
qty += cache[d]
return qty
def solve():
factors = [0] * 20
accum = 1
sieve(20)
for i in xrange(1, 21):
for p,k in factor(i):
if k > factors[p-1]:
accum *= p ** (k - factors[p-1])
factors[p-1] = k
return accum
def solve():
sieve(10**6)
mdrs = [None] * 10**6
mdrs[0] = 0 # to emulate the omission of 1 from factorizations
accum = 0
for n in xrange(2, 10**6):
val = max(droot(d) + mdrs[n // d - 1] for d in divisors(n) if d != 1)
accum += val
mdrs[n-1] = val
return accum
def solve():
n = 28123
sieve(n+1)
summable = set()
abundant = []
for i in xrange(1, n+1):
if aliquot(i) > i:
abundant.append(i)
summable.update(i+j for j in abundant if i+j <= n)
return n*(n+1) // 2 - sum(summable)
def solve():
sieve(10000)
cache = {}
accum = 0
for i in xrange(2, 10001):
j = aliquot(i)
if j < i and cache.get(j) == i:
accum += i + j
elif j > i:
cache[i] = j
return accum
def solve():
sieve(1000)
qty = 2 # 2 and 5
for p in xrange(3, 1000000, 2):
pstr = str(p)
if (
all(c in "1379" for c in pstr)
and isPrime(p, presieve=False)
and all(isPrime(int(pstr[i:] + pstr[:i]), presieve=False) for i in range(1, len(pstr)))
):
qty += 1
return qty
def solve():
sieve(10**7)
new = [(d,d) for d in range(1, 10)]
accum = 0
for _ in xrange(12):
newer = []
for n,s in new:
for d in xrange(10):
n2 = n*10 + d
s2 = s + d
q,r = divmod(n2, s2)
if r == 0:
newer.append((n2, s2))
if isPrime(q, presieve=False):
for e in [1,3,7,9]:
m = n2 * 10 + e
if isPrime(m, presieve=False):
accum += m
new = newer
return accum
import sys
from eulerlib import sieve
P = sieve(1000000)
S = set(P)
def rotate(s, I):
d = 0
for n in xrange(1 if 0 in I else 0, 10):
C = [c for c in s]
for i in I:
C[i] = str(n)
if int(''.join(C)) in S:
d += 1
return d
for p in P:
s = str(p)
for d in set(s):
I = []
for i in xrange(len(s)):
v = -1
if s[i] == d:
I.append(i)
v = rotate(s, I)
if v == 8:
print s
found = True
sys.exit(0)
from eulerlib import sieve
N = 10000
M = 100000000
primes = sieve(N)
prime_set = set(sieve(M))
def concat(a, b):
return int(str(a) + str(b))
def get_concat_sets(n):
if n == 0:
return []
elif n == 1:
return [[n] for n in primes]
concat_sets = []
prev_sets = get_concat_sets(n - 1)
for s in prev_sets:
for p1 in primes[primes.index(max(s)) + 1:]:
found = True
for p2 in s:
if concat(p1, p2) not in prime_set or concat(
p2, p1) not in prime_set:
found = False
break
if found:
concat_sets.append(s + [p1])
from eulerlib import sieve N = 1000000 P = sieve(N) S = set(P) m = -1 p = -1 for i in xrange(len(P)): s = 0 j = i while s < N and j < len(P): s += P[j] j += 1 if s in S and j-i > m: m = j-i p = s print p
import sys; sys.path.insert(1, sys.path[0] + '/..')
from eulerlib import factor, modInverse, cross, sieve
def I(n):
terms = []
for (p,k) in factor(n):
if p % 2 == 0 and k>1:
residues = (1, -1, (1<<k-1)-1, (1<<k-1)+1)
else:
residues = (1, -1)
pk = p**k
coef = n // pk * modInverse(n // pk, pk)
terms.append(map(lambda x: x*coef, residues))
return max(filter(lambda x: x != n-1, (sum(res) % n for res in cross(*terms))))
sieve(2 * 10**7)
#print sum(I(i) for i in xrange(3, 20000001))
for i in xrange(3, 20000001):
print '%d\t%d' % (i, I(i))
from eulerlib import sieve N = 10000 M = 100000000 primes = sieve(N) prime_set = set(sieve(M)) def concat(a,b): return int(str(a)+str(b)) def get_concat_sets(n): if n == 0: return [] elif n == 1: return [[n] for n in primes] concat_sets = [] prev_sets = get_concat_sets(n-1) for s in prev_sets: for p1 in primes[primes.index(max(s))+1:]: found = True for p2 in s: if concat(p1,p2) not in prime_set or concat(p2,p1) not in prime_set: found = False break if found: concat_sets.append(s + [p1]) return concat_sets
def solve():
sieve(1000000)
return sum(totient(n) for n in xrange(2, 1000001))
import sys
from eulerlib import sieve
P = sieve(1000000)
S = set(P)
def rotate(s,I):
d = 0
for n in xrange(1 if 0 in I else 0, 10):
C = [c for c in s]
for i in I:
C[i] = str(n)
if int(''.join(C)) in S:
d += 1
return d
for p in P:
s = str(p)
for d in set(s):
I = []
for i in xrange(len(s)):
v = -1
if s[i] == d:
I.append(i)
v = rotate(s,I)
if v == 8:
print s
found = True
sys.exit(0)
from eulerlib import sieve
N = 100000
S = [2 * n * n for n in xrange(N)]
P = set(sieve(N))
C = [n for n in xrange(9, N, 2) if n not in P]
for n in C:
satisfied = False
for s in S:
if n - s in P:
satisfied = True
break
if not satisfied:
print n
break
def solve():
sieve(10 ** 6) # just a guess
return sum(islice(answers(), 25))
from eulerlib import sieve, cmpf
N = 1000000
prime_factors = [[], [(1, 1)]] + [[] for n in xrange(N - 1)]
phi = [0, 1] + [-1 for n in xrange(N - 1)]
all_primes = sieve(N)
sub_primes = [p for p in all_primes if p <= N / 2]
#calculate prime factors
for p in sub_primes:
n = p
while n <= N:
m = n
s = 0
while m % p == 0:
s += 1
m /= p
prime_factors[n].append((p, s))
n += p
#calculate Euler Phi function
for p in all_primes:
n = p
prev = phi[1]
while n <= N:
phi[n] = n - n / p
prev = n
n *= p
from eulerlib import sieve P = set(sieve(10000)) def isperm(a,b): s1 = set(str(a)) s2 = set(str(b)) return s1 == s2 for n0 in xrange(1000,10000): if n0 not in P: continue for i in xrange(1,10000): n1 = n0 + i n2 = n1 + i if n1 > 9999 or n2 > 9999: break if n1 in P and n2 in P and isperm(n0,n1) and isperm(n0,n2): print n0,n1,n2,i
def solve():
n = 10001
sieve(ceil(n*log(n) + n*log(log(n))))
return nth(primeIter(), n-1)
from eulerlib import sieve N = 100000 S = [2*n*n for n in xrange(N)] P = set(sieve(N)) C = [n for n in xrange(9,N,2) if n not in P] for n in C: satisfied = False for s in S: if n-s in P: satisfied = True break if not satisfied: print n break
from eulerlib import sieve, cmpf N = 1000000 prime_factors = [[],[(1,1)]] + [[] for n in xrange(N-1)] phi = [0,1] + [-1 for n in xrange(N-1)] all_primes = sieve(N) sub_primes = [p for p in all_primes if p <= N/2] #calculate prime factors for p in sub_primes: n = p while n <= N: m = n s = 0 while m % p == 0: s += 1 m /= p prime_factors[n].append((p,s)) n += p #calculate Euler Phi function for p in all_primes: n = p prev = phi[1] while n <= N: phi[n] = n - n/p prev = n n *= p
from eulerlib import sieve
P = set(sieve(1000000))
def iscircular(n):
s = str(n)
for i in xrange(len(s)):
if int(s[i:] + s[:i]) not in P:
return False
return True
N = [n for n in xrange(1000000) if iscircular(n)]
print len(N)
def solve():
sieve(10**8) # A 9-pandigital cannot be prime.
return primeSets(set("123456789"))
