ans = 1
for i in range(1, modulus - n):
ans = (ans * i) % modulus
ans = gmpy2.invert(ans, modulus)
# Since m is an odd-prime, (-1)^(m-n) = -1 if n is even, +1 if n is odd
if n % 2 == 0:
ans = -1 * ans + modulus
return ans % modulus
t = time()
print "calculating primes up to the 8th power"
p = primes(10 ** 8)
print "primes generated"
summ = 0
p.pop(0)
p.pop(0)
for x in p:
z = x - 1 # factorialMod(x-1,x)
z1 = 1 # factorialMod(x-2,x)
z2 = z / 2 # factorialMod(x-3,x)
z3 = factorialMod(x - 4, x)
z4 = factorialMod(x - 5, x)
# print x,":",z,z1,z2,z3,z4
z += z1 + z2 + z3 + z4
n += 1
f = defaultdict(list)
for p in xrange(2, n):
if p not in f:
for i in xrange(p + p, n, p):
j, k = i, 1
while j % p == 0:
j //= p
k *= p
f[i].append(p)
if f[p]==[]:f[p]=[p]
return f
p = primes(int((10**18/4)**(1/3.)))
F = FactorSieve(p[-1])
c = len(p)
plist={}
for i in xrange(2,200):
q = p[i]
l = [q]
l2 = []
f = F[q-1]
for z in f:
f1 = F[z-1]
for z2 in f1:
#
#
# Euler 278
#
#
from time import time
from Functions import primes
st = time()
t = 0
N = 5000
ps = primes(N)
C = []
for p in ps:
t += p
C.append(t)
#print C
t = 0
for j in xrange(1, len(ps) - 1):
q = ps[j]
t += (2 * q - 1) * C[j - 1] * (C[-1] - C[j]) - q * (
C[-1] - C[j]) * j - q * (C[j - 1]) * (len(ps) - j - 1)
print "sum of f(a,b,c) is ", t
print "time elapsed ", time() - st
#
# Euler 204
#
from Functions import RetFact, IsPrime, primes
pns = primes(10**9)
summ = 0
for i in xrange(2, 10**9):
if i > 100 and not (i in pns): continue
zz = sorted(RetFact(i), reverse=True)
if zz[0] < 101:
print i, zz[0]
summ += 1
print
print "Hamming Number type 100 <10^9 is", summ + 1
return False
return True
def cntpowers(x):
z = RetFact(x)
S = set(z)
summ = 0
for SS in S:
summ += z.count(SS)
return summ
N = 4 * 10**5
p = primes(N)
X = 5
summm = 0
Z = 60
for i in xrange(Z, Z + 1):
if i in p:
summm += 1
continue
d = divisors(i)
D = cntpowers(i)
D2 = D / 2
print i, d, D, D2, RetFact(i)
for x in d:
q = cntpowers(x)
if q == D2:
if z/y==(1.0 * z)/y:
return False
return True
def cntpowers(x):
z = RetFact(x)
S = set(z)
summ = 0
for SS in S:
summ += z.count(SS)
return summ
N = 4*10**5
p = primes(N)
X=5
summm=0
Z=60
for i in xrange(Z,Z+1):
if i in p:
summm+=1
continue
d = divisors(i)
D = cntpowers(i)
D2=D/2
print i,d,D,D2,RetFact(i)
for x in d:
q = cntpowers(x)
if q == D2:
#
#
# Euler 278
#
#
from time import time
from Functions import primes
st=time()
N = 5000
summ = 0
pr=primes(N)
l = len(pr)
for i in xrange(l-2):
p = pr[i]
for j in xrange(i+1,l-1):
q = pr[j]
pq = p*q
peasnqueues = 2*pq - p - q
for k in xrange(j+1,l):
summ += pr[k]*(peasnqueues) -pq
print "sum of f(a,b,c) is ", summ
print "time elapsed ", time()-st
# # # Euler 278 # # from time import time from Functions import primes st=time() t=0 N=5000 ps=primes(N) C=[] for p in ps: t+=p C.append(t) #print C t=0 for j in xrange(1,len(ps)-1): q=ps[j] t+=(2*q-1)*C[j-1]*(C[-1]-C[j])-q*(C[-1]-C[j])*j-q*(C[j-1])*(len(ps)-j-1) print "sum of f(a,b,c) is ", t print "time elapsed ", time()-st
from Functions import gcd, RetFact, divisors
from math import factorial
from Functions import primes
from collections import defaultdict
from time import time
from itertools import combinations
st = time()
p = primes(10**8)
print "time to calculate seive up to 10**8 is ", time()-st
print len(p)
mx = (10**8)/2
d=defaultdict(int)
tot = 1
for i in xrange(2,10**8+1):
if i in p:
d[i]=1
else:
f = RetFact(i)
for ff in f:
d[ff]+=1
# for dd in d:
# print dd,d[dd]
# l = sorted(d)
for d in D:
tot += d * d
return tot
st = time()
maxp = 44
# res = 15015
# hasRes = [False]* res
# for i in xrange(res):
# hasRed[i*i % res] = True
p = primes(maxp)
total = 1
for i in xrange(maxp + 1): #00000):
if i in p: continue
o = O2(i)
if o**.5 == int(o**.5):
print i, o, divisors(i), RetFact(i)
print
total += i
print "Sum of O2 for i<64,000,000 is", total
print "process time", time() - st
#
#
# Euler 278
#
#
from time import time
from Functions import primes
st = time()
N = 5000
summ = 0
pr = primes(N)
l = len(pr)
for i in xrange(l - 2):
p = pr[i]
for j in xrange(i + 1, l - 1):
q = pr[j]
pq = p * q
peasnqueues = 2 * pq - p - q
for k in xrange(j + 1, l):
summ += pr[k] * (peasnqueues) - pq
print "sum of f(a,b,c) is ", summ
print "time elapsed ", time() - st
s,s0=0,1
t,t0=1,0
r,r0=b,a
while r != 0:
q = r0/r
r0,r = r,r0-q*r
s0,s = s,s0-q*s
t0,t = t,t0-q*t
# print q,r,r0,s,s0,t,t0
# print "bezouts coeffs", s0,t0
# print "gcd ", r0
# print "quots by the gcd", t,s
return s0,t0,r0
st=time()
p = primes(1000010)
l = len(p)
summ=0
for i in xrange(2,l-1):
j = i+1
v1 = p[i]
l1 = len(str(v1))
base = 10**l1
inc = 1
# print v1, base,inc, p[j]
delta=p[j]-v1
v2=p[j]
def gm(a, b, c):
if g(a, b) > 1: return False
#if g(b,c)>1:return False
#if g(a,c)>1:return False
return True
st = time()
N = 1000
ctr = 0
print "Calculating Factor Sieve for N=", N, "...",
f = FactorSieve(N)
print "done."
l = [1]
p = primes(620)
l += p
print "Generating list of semi-primes and powers...",
for i in xrange(2, N):
s0 = set(f[i])
if (len(s0) == 1
and len(f[i]) > 1) or (len(s0) == 2
and len(f[i]) > 2) or (len(s0) == 3
and len(f[i]) > 3):
l.append(i)
l = sorted(l)
print "done."
# for x in l:
# print x,
# exit()
for i in xrange(p + p, n, p):
j = i
while j % p == 0:
j //= p
f[i].append(p)
#f[i]=[p]
#break
if p not in f: f[p] = [p]
return f
st = time()
F = FactorSieve(5000000)
p = primes(20000000)
#print F
print time() - st
#exit()
summ = 0
#x=20000000
#y=5000000
z = 5000000
summ += 4 * 1250000 # extract every 4th numerator with first 1250000 factors of 5000000! 20,000,000/5,000,000 = 4 , etc
for x in xrange(19999998, 19900000, -4):
# if x in F:
# v=sum(F[x])
if g(a,b)==g(b,c)==g(a,c)==1:return True
def gm(a,b,c):
if g(a,b)>1:return False
#if g(b,c)>1:return False
#if g(a,c)>1:return False
return True
st=time()
N=1000
ctr=0
print "Calculating Factor Sieve for N=",N,"...",
f = FactorSieve(N)
print "done."
l = [1]
p=primes(620)
l+=p
print "Generating list of semi-primes and powers...",
for i in xrange(2,N):
s0=set(f[i])
if (len(s0)==1 and len(f[i])>1) or (len(s0)==2 and len(f[i])>2) or (len(s0)==3 and len(f[i])>3):l.append(i)
l = sorted(l)
print "done."
# for x in l:
# print x,
# exit()
for c in l:
for a in l:
s1=set(f[c])
#if len(s1)>2:continue
#
# Euler 204
#
from Functions import RetFact,IsPrime,primes
pns = primes(10**9)
summ = 0
for i in xrange(2,10**9):
if i> 100 and not (i in pns):continue
zz = sorted(RetFact(i),reverse=True)
if zz[0]<101:
print i,zz[0]
summ+=1
print
print "Hamming Number type 100 <10^9 is",summ+1
hasOdds = False
for zz in xrange(1,len(psetlist)):
if gcd(psetlist[zz],psetlist[zz-1])==1:
hasOdds=True
break
if hasOdds==False:return False
return True
l=[5,3]
N=864
Lim=10**8
#p,q=2,3
p =1217 #,q=2,8191
ctr=0
qp=primes(600000)
#F=FactorSieve(2000000)
#for q in qp: len(qp)-1000
sctr=0
for qq in xrange(1,len(qp)):
q = qp[qq]
ii,jj,vv=0,0,0
imax,jmax=0,0
ctr=0
for i in xrange(2,56): #57
for j in xrange(3,36):
if gcd(i,j)>1:continue
n = [p]*i + [q]*j
v = reduce(mul,n)
j = i
while j % p == 0:
j //= p
# f[i].append(p)
f[i] = [p]
break
if p not in f:
f[p] = [p]
return f
st = time()
F = FactorSieve(500000)
p = primes(20000000)
# print F
print time() - st
# exit()
summ = 0
# x=20000000
# y=5000000
z = 5000000
for x in xrange(20000000, 20000000 - 100, -1):
if x in p:
summ += x
print "p", x
continue
def A(n):
k = 2
while True:
r = R(k)
z = r / n
#print n,k,r,z
if n * z == r:
return k
k += 1
st = time()
p = primes(100000)
print "time to gen primes", time() - st
summ = 0
ctr = 0
for i in xrange(11, 100000, 2):
if i in p: continue
if gcd(i, 10) > 1: continue
v = A(i)
w = i - 1
z = w / v
if z * v == w:
print i
ctr += 1
summ += i
if ctr == 25: break
def O2(l):
tot = 0
D = divisors(n)
#print n,D
for d in D:
tot += d*d
return tot
st = time()
maxp = int(64000000**.5)
p=primes(maxp)
print "Prime Max", p[-1]
total = 0
for i in p:
for j in p:
z = (i,j)
sigma2=O2(z)
if int(x**.5)**2 == x:
print "n,x:", i*j,i,j,x
print "Sum of O2 for i<64,000,000 is",total
print "process time",time()-st
if f[p] == []: f[p] = [p]
return f, t
def phi(n):
v = n
f = list(set(RetFact(n)))
for z in f:
v *= (z - 1)
v /= z
return v
st = time()
p = primes(40000000)
F, Phi = Factor(10000001)
print time() - st
#print len(Phi)
def chain(n, Phi, p):
dispch = False
if n == 9548417 or n == 9576857 or n == 9577397:
dispch = True
ctr = 1
q = r0/r
r0,r = r,r0-q*r
s0,s = s,s0-q*s
t0,t = t,t0-q*t
# print q,r,r0,s,s0,t,t0
# print "bezouts coeffs", s0,t0
# print "gcd ", r0
# print "quots by the gcd", t,s
return r0
print ext_gcd(240,46)
exit()
p = primes(1000000)
l = len(p)
summ=0
for i in xrange(2,l-1):
j = i+1
v1 = p[i]
l1 = len(str(v1))
base = 10**l1
inc = 1
# print v1, base,inc, p[j]
while True:
ctr=base*inc
if f[p]==[]:f[p]=[p]
return f, t
def phi(n):
v = n
f = list(set(RetFact(n)))
for z in f:
v *= (z-1)
v /= z
return v
st = time()
p=primes(40000000)
F,Phi=Factor(10000001)
print time()-st
#print len(Phi)
def chain(n,Phi,p):
dispch=False
if n==9548417 or n==9576857 or n==9577397:
dispch=True
ctr=1