def p141():
L = []
s, limita, limitm = 0, 10**4, 10**12
for a in xrange(2, limita):
for b in xrange(1, a):
if (a**3) * b + (b**2) >= limitm:
break
if gcd(a, b) > 1:
continue
c = 1
while True:
m2 = (c**2) * (a**3) * b + c * (b**2)
if m2 > limitm:
break
else:
if m2**.5 == int(m2**.5):
s += m2
L.append(m2)
c += 1
print 'Solution: {}'.format(s)
L = sorted(L)
for m2 in L:
print m2, RetFact(m2)
def p141():
L = []
s, limita, limitm = 0, 10**4, 10**12
for a in xrange(2,limita):
for b in xrange(1,a):
if (a**3)*b+(b**2)>=limitm:
break
if gcd(a,b)>1:
continue
c=1
while True:
m2=(c**2)*(a**3)*b+c*(b**2)
if m2>limitm:
break
else:
if m2**.5==int(m2**.5):
s+=m2
L.append(m2)
c+=1
print 'Solution: {}'.format(s)
L=sorted(L)
for m2 in L:
print m2, RetFact(m2)
def gcd2(l):
bzzz = True
for z in xrange(len(l)):
for y in xrange(len(l)):
if z==y:continue
if gcd(l[z],l[y])>1:
bzzz = False
break
return bzzz
def gcd2(l):
bzzz = True
for z in xrange(len(l)):
for y in xrange(len(l)):
if z == y: continue
if gcd(l[z], l[y]) > 1:
bzzz = False
break
return bzzz
def R(M):
tot = 0.0
c = int(ceil(M/2))
for i in xrange(1,M):
start = M-i
for j in xrange(start,M+1):
#if i == j: continue
s = i+j
if gcd(i,j)!=1:continue
#print "M:",M,i,j,s
if s>=M:
print "M:",M,i,j,s
y = 1.0/(i*j)
tot +=y
return tot
def R(M):
tot = 0.0
c = int(ceil(M / 2))
for i in xrange(1, M):
start = M - i
for j in xrange(start, M + 1):
#if i == j: continue
s = i + j
if gcd(i, j) != 1: continue
#print "M:",M,i,j,s
if s >= M:
print "M:", M, i, j, s
y = 1.0 / (i * j)
tot += y
return tot
def R(M):
tot = 0.0
ctr = 0
#c = int(ceil(M/2))
for i in xrange(1,M):
#start = M-i
for j in xrange(i+1,M+1):
#if i == j: continue
if gcd(i,j)!=1:continue
s = i+j
#print "M:",M,i,j,s
if s>=M:
print "M:",M,i,j,s
y = 1.0/(i*j)
tot +=y
ctr +=1
return tot,ctr
def R(M):
tot = 0.0
ctr = 0
#c = int(ceil(M/2))
for i in xrange(1, M):
#start = M-i
for j in xrange(i + 1, M + 1):
#if i == j: continue
if gcd(i, j) != 1: continue
s = i + j
#print "M:",M,i,j,s
if s >= M:
print "M:", M, i, j, s
y = 1.0 / (i * j)
tot += y
ctr += 1
return tot, ctr
def p309(N = 10**6):
lll = [[] for w in xrange(N)]
for s in xrange(1, int(N**0.5) + 500):
for t in xrange(1 + (s&1), s, 2):
if s*s + t*t >= N:
break
if gcd(s, t) == 1:
for l in xrange(1, N/(s*s + t*t) + 1 - ((N%(s*s + t*t)) == 0)):
a = 2*l*s*t
b = l*(s*s - t*t)
lll[a].append(b)
lll[b].append(a)
count = 0
for ll in lll:
for i, a in enumerate(ll):
for b in ll[:i]:
if (a*b) % (a+b) == 0:
print a,b
count += 1
return count
def isAchilles(l):
s = set(l)
prev = 0
psetlist=[]
#hasOdds=False
for v in s:
z = l.count(v)
if z < 2: return False
#if z%2==1: hasOdds=True
psetlist.append(z)
if len(set(psetlist))==1:
return False
#if hasOdds==False:
# return False
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
def isAchilles(l):
s = set(l)
prev = 0
psetlist=[]
#hasOdds=False
for v in s:
z = l.count(v)
if z < 2: return False
#if z%2==1: hasOdds=True
psetlist.append(z)
if len(set(psetlist))==1:
return False
#if hasOdds==False:
# return False
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
def p309(N=10**6):
lll = [[] for w in xrange(N)]
for s in xrange(1, int(N**0.5) + 500):
for t in xrange(1 + (s & 1), s, 2):
if s * s + t * t >= N:
break
if gcd(s, t) == 1:
for l in xrange(
1, N / (s * s + t * t) + 1 - ((N %
(s * s + t * t)) == 0)):
a = 2 * l * s * t
b = l * (s * s - t * t)
lll[a].append(b)
lll[b].append(a)
count = 0
for ll in lll:
for i, a in enumerate(ll):
for b in ll[:i]:
if (a * b) % (a + b) == 0:
print a, b
count += 1
return count
def euler283(low=1,top=1000):
M=[]
st = time()
t=0
for m in xrange(low,top+1):
print "m:",m,
psum = 0
for u in xrange(1,2*m+1):
if (2*m)%u: continue
for v in xrange(1,int(floor(sqrt(3)*u)+1)):
if gcd(v,u)!=1: continue
F=4*m**2*(u**2+v**2)
#fs=set(RetFact(m))
#fs=fs.union(set(RetFact(u**2+v**2)))
#fs.add(2)
for d1 in divisors(F):
if (d1+2*m*u)%v: continue
d2=F/d1
if d1>d2: continue
if (d2+2*m*u)%v: continue
a=(d1+2*m*u)/v + 2*m*v/u
b=(d2+2*m*u)/v + 2*m*v/u
c=(d1+d2+4*m*u)/v
if b>c: continue
#print m,":",a,b,c,a+b+c,d1,d2
t+=a+b+c
psum+=a+b+c
print psum,
print
M.append(psum)
print t
print time()-st
print
print
print M
#
#
from time import time
from math import floor
from Functions import divisors,gcd
st=time()
N = 10**8
limit = int(N**.5)
#ctr = 1
summ=0
summ2=0
for a in xrange(1,limit+1):
for b in xrange(a,limit+1):
if gcd(a,b)!=1:continue
#a2,b2=pow(a,2),pow(b,2)
a2,b2=a**2,b**2
div = a2 + b2
if div>N:break
#ctr+=1
#No need for floor func. Runs slow, Int div in Python defaults to floor
#Np=floor(N/div)
if a==b:
z=(a<<1)
else:
z=(a+b)<<1
# num0=div=(a^2+b^2) -> (num0, 2num0, 3num0....n*num0)/(div) == complex divisors
# then take multiples of div, recalc floor and sum again
for n in xrange(1, N/div +1):
#
from Functions import gcd
'''
A=[5248,1312,2624,5760,3936]
B=[640,1888,3776,3776,5664]
u0=1476;v0=1475
'''
LIMIT = 1000
i, j, k, t, t2, g, s, s2 = 0, 0, 0, 0, 0, 0, 0, 0
for i in xrange(1, LIMIT + 1):
for j in xrange(1, LIMIT + 1):
for k in xrange(1, LIMIT + 1):
t = 6372 * (41 * i + 205 * j + 5 * k)
t2 = 205 * (90 * i + 5310 * j + 59 * k)
g = gcd(t, t2)
t /= g
t2 /= g
s = int(t**.5)
if s * s == t:
s2 = int(t2**.5)
if (s2 * s2 == t2 and s * 41 > 59 * s2 and s * i <= 37760
and s * j <= 11328 and s * k <= 339840):
print i, j, k, s, s2
ctr=0
for d in range(5,D + 1):
if d in primes: # IsPrime(d):
strt = ceil(d*mn)
endr = floor(d*mx)
ctr += endr - strt + 1
else:
#facts = RetFact(d)
strt = ceil(d*mn)
endr = floor(d*mx)
#print "Comp D:",d,strt,endr+1
#g=RetFact(d)
for i in xrange(int(strt),int(endr+1)):
#a = RetFact(i)
#if bool(set(a) & set(g))==False:
#print i,":",a
if gcd(i,d) ==1:
ctr+=1
#print d,RetFact(i)
#else:
# print d,RetFact(i)
print "Answer is",ctr
print "Process time is", time()-st
#short sol
#len(set([float(a)/b for a in xrange(1,12001) for b in range(a+1,12001) if float(a)/b > 1.0/3 if float(a)/b < .5 ]))
D = 12000
f = RetFact(D)
ctr = 0
for d in range(5, D + 1):
if d in primes: # IsPrime(d):
strt = ceil(d * mn)
endr = floor(d * mx)
ctr += endr - strt + 1
else:
#facts = RetFact(d)
strt = ceil(d * mn)
endr = floor(d * mx)
#print "Comp D:",d,strt,endr+1
#g=RetFact(d)
for i in xrange(int(strt), int(endr + 1)):
#a = RetFact(i)
#if bool(set(a) & set(g))==False:
#print i,":",a
if gcd(i, d) == 1:
ctr += 1
#print d,RetFact(i)
#else:
# print d,RetFact(i)
print "Answer is", ctr
print "Process time is", time() - st
#short sol
#len(set([float(a)/b for a in xrange(1,12001) for b in range(a+1,12001) if float(a)/b > 1.0/3 if float(a)/b < .5 ]))
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
print "Sum,ctr = ", summ, ctr
print "time elapsed is", time() - st
from Functions import gcd, RetFact, divisors
from math import factorial
from Functions import primes
# p = primes(10**8)
# print len(p)
# exit()
for i in xrange(7, 8):
n = factorial(i)
div = divisors(n)
t = 0
sqrtn = int(n**.5)
print div
for j in div:
#if j> sqrtn:break
if gcd(j, n / j) == 1:
f = RetFact(j)
#if set(f)==set([2,3,5,7]):
print i, n, len(f), f, j
t += j**2 # + (n/j)**2
# print "tot:",i,n,RetFact(t), t%1000000009
# print "tot:",i,RetFact(i),n, t, t%1000000009,RetFact(t%1000000009)
# print "tot:",i,n,RetFact(n), t,RetFact(t), t%1000000009
print
#
#
from Functions import gcd
Lim = 10000 #100000000
d = {}
sum = 0
z = 0
for i in xrange(1, Lim + 1):
sum += (Lim / i) * i
for a in xrange(1, int(Lim**.5) + 1):
for b in xrange(1, a + 1): # /*&& (a*a+b*b)<=Lim*/;b++)
if (a, b) in d:
z = d[(a, b)]
else:
d[(a, b)] = gcd(a, b)
if (z == 1):
i = (a * a + b * b)
if (a == b):
val = (a + b)
else:
val = 2 * (a + b)
for j in xrange(1, int(Lim / i) + 1):
sum += (j * val * (Lim / (i * j)))
print sum
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)
if v> 10**18: break
A= isAchilles(n)
if not A:continue
phiA = Phi(v,n)
fphi=RetFact(phiA) #F[phiA]
#fphi=F[phiA]
B = isAchilles(fphi)
if not B:continue
ctr+=1
if v>vv:ii,jj,vv=i,j,v
if i>imax:imax=i
if j>jmax:jmax=j
print i,j,":" , v, fphi
#
#
# Euler Problem 75
#
#
from Functions import gcd
from time import time
from math import sqrt
st = time()
lim = 1500000
lim2 = int(sqrt(lim))
ar = [0] * (lim + 1)
for m in xrange(1, lim2, 2):
for n in xrange(2, lim2 - m, 2):
if gcd(m, n) == 1:
#print "m,n",m,n
a = abs(m**2 - n**2)
b = 2 * m * n
c = m**2 + n**2
l = a + b + c
#k=1
for s in range(l, lim, l):
ar[s] += 1
'''
while l<=lim:
#l *= k
#if l>lim:break
ar[l]+=1
l+=l
return s_local + s_smooth
#-------------------------------
# Complex divisors
#-------------------------------
def g_sum(n, p):
'''Return the inner-most sum of complex Gaussian divisors with positive real part of all k, k=1..n.
Using a summation formula. Runtime complexity: O(n^(1/2)).'''
# Optimal split to balance local and smoth work
q, s = int((n / p) ** 0.5), 0L
# Local part
for g in xrange(1L, n / (p * q) + 1):
K = n / (p * g)
s += (K * g + K * (K + 1) / 2)
# Smooth part
for k in xrange(1L, q):
r1, r2 = n / (p * (k + 1)), n / (p * k)
G = (r1 + r2 + 1) * (r2 - r1) / 2
s += (k * G + k * (k + 1) * (r2 - r1) / 2)
return s
'''Return the sum of complex Gaussian divisors with positive real part of all k, k=1..n.
Using a summation formula. Runtime complexity: O(n).'''
sum_divisors_complex = lambda n: \
sum(a * sum(g_sum(n, a * a + b * b) for b in xrange(1, int((n - a * a) ** 0.5) + 1) if gcd(a, b) == 1)
for a in xrange(1, int((n - 1) ** 0.5) + 1))
sum_divisors_gaussian = lambda n: sum_divisors_rational(n) + sum_divisors_complex(n)
if __name__ == "__main__":
print sum_divisors_gaussian(10L ** 8)
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
print "Sum,ctr = ", summ, ctr
print "time elapsed is", time() - st
#
#
#
from Functions import gcd
def A(n):
curRepUnit = 1
k = 0
ctr = 1
while (curRepUnit % n != 0):
curRepUnit = pow(10, ctr) / 9
k += 1
ctr += 1
return k
##########################################
for n in xrange(1000020, 1000200):
if gcd(n, 10) != 1: continue
print n, A(n)
from Functions import gcd, RetFact, divisors
from math import factorial
from Functions import primes
# p = primes(10**8)
# print len(p)
# exit()
for i in xrange( 7 , 8):
n = factorial(i)
div = divisors(n)
t = 0
sqrtn = int(n**.5)
print div
for j in div:
#if j> sqrtn:break
if gcd(j,n/j) == 1:
f = RetFact(j)
#if set(f)==set([2,3,5,7]):
print i, n, len(f),f, j
t += j**2 # + (n/j)**2
# print "tot:",i,n,RetFact(t), t%1000000009
# print "tot:",i,RetFact(i),n, t, t%1000000009,RetFact(t%1000000009)
# print "tot:",i,n,RetFact(n), t,RetFact(t), t%1000000009
print
def totient(n):
tot, pos = 0, n-1
while pos>0:
if gcd(pos,n)==1: tot += 1
pos -= 1
return tot
from Functions import gcd
from math import sqrt
def primitive(a, b, c):
if a%2 or b%2 or c%2:
return (a, b, c)
else:
primitive(a, b, c)
opt = []
count = 0
print 'Lijst met alle mogelijke rechthoekige 3hoeken genereren'
for m in xrange(1, int(sqrt(0.5 * 15 * 10**5))):
for n in xrange(1, m):
if gcd(m, n) == 1 and (m - n) % 2:
L = 2 * m * (m + n)
for k in xrange(1, (15 * 10**5) // L + 1):
L = k * 2 * m * (m + n)
# a = m**2 - n**2
# b = 2*m*n
# c = m**2 + n**2
# if a > b:
# a, b = b, a
# print a, b, c
if L < 15 * 10**5:
opt.append(L)
# print L, m, n, k
print 'Lijst gegenereerd ({0} items/{1} distinct), checken op multipliciteit'.format(len(opt), len(set(opt)))
# Euler Problem 75
#
#
from Functions import gcd
from time import time
from math import sqrt
st = time()
lim = 1500000
lim2 = int(sqrt(lim))
ar = [0] * (lim + 1)
for m in xrange(1, lim2, 2):
for n in xrange(2, lim2 - m, 2):
if gcd(m, n) == 1:
# print "m,n",m,n
a = abs(m ** 2 - n ** 2)
b = 2 * m * n
c = m ** 2 + n ** 2
l = a + b + c
# k=1
for s in range(l, lim, l):
ar[s] += 1
"""
while l<=lim:
#l *= k
#if l>lim:break
ar[l]+=1
#
#
#
#
from Functions import gcd
from time import time
summ=0;
Lim=100000
st=time()
for i in xrange(1,Lim+1): # for(int i=1;i<=Lim;i++)
summ+=(Lim/i)*i
for a in xrange(1,int(Lim**.5)+1): #(int a=1;a*a<Lim;a++)
for b in xrange(1,a+1): #(int b=1;b<=a /*&& (a*a+b*b)<=Lim*/;b++)
if (gcd(a,b)==1):
div=(a*a+b*b)
if (a==b):
val = (a+b)
else:
val=2*(a+b);
for j in xrange(1, Lim/div +1): #(int j=1;i*j<=Lim;j++) :
summ+=(j*val*(Lim/(div*j)))
print summ
print time()-st
#
#
#
#
from Functions import gcd
from time import time
summ = 0
Lim = 100000
st = time()
for i in xrange(1, Lim + 1): # for(int i=1;i<=Lim;i++)
summ += (Lim / i) * i
for a in xrange(1, int(Lim**.5) + 1): #(int a=1;a*a<Lim;a++)
for b in xrange(1, a + 1): #(int b=1;b<=a /*&& (a*a+b*b)<=Lim*/;b++)
if (gcd(a, b) == 1):
div = (a * a + b * b)
if (a == b):
val = (a + b)
else:
val = 2 * (a + b)
for j in xrange(1, Lim / div + 1): #(int j=1;i*j<=Lim;j++) :
summ += (j * val * (Lim / (div * j)))
print summ
print time() - st
def getDigitCount(n):
# the first number to check is 1
current = 1
# 1 has a digit count of 1
digitCount = 1
# while n is not divisible by the current number
while (current % n != 0):
#print "current",current,current % n
# we add one 1 to the end of current
current = (current * 10 + 1) % n
#and increment the digit count
digitCount += 1
return digitCount
##########################################
st = time()
for i in xrange(1000020, 1000030):
if gcd(i, 10) != 1: continue
print i, getDigitCount(i)
print "time elapsed is ", time() - st
#
#
from Functions import gcd
Lim = 10000 #100000000
d={}
sum=0
z=0
for i in xrange(1,Lim+1):
sum+=(Lim/i)*i;
for a in xrange(1,int(Lim**.5)+1):
for b in xrange(1,a+1): # /*&& (a*a+b*b)<=Lim*/;b++)
if (a,b) in d:
z = d[(a,b)]
else:
d[(a,b)]=gcd(a,b)
if(z==1):
i=(a*a+b*b)
if (a==b):
val=(a+b)
else:
val=2*(a+b);
for j in xrange(1,int(Lim/i)+1):
sum+=(j*val*(Lim/(i*j)));
print sum
def getDigitCount(n):
# the first number to check is 1
current = 1;
# 1 has a digit count of 1
digitCount = 1;
# while n is not divisible by the current number
while (current % n != 0):
#print "current",current,current % n
# we add one 1 to the end of current
current = (current * 10 + 1)%n
#and increment the digit count
digitCount+=1
return digitCount
##########################################
st = time()
for i in xrange(1000020,1000030):
if gcd(i,10)!=1:continue
print i, getDigitCount(i)
print "time elapsed is ", time()-st
#
from Functions import gcd
'''
A=[5248,1312,2624,5760,3936]
B=[640,1888,3776,3776,5664]
u0=1476;v0=1475
'''
LIMIT = 1000
i,j,k,t,t2,g,s,s2=0,0,0,0,0,0,0,0
for i in xrange(1,LIMIT+1):
for j in xrange(1,LIMIT+1):
for k in xrange(1,LIMIT+1):
t = 6372*(41*i+205*j+5*k)
t2 = 205*(90*i+5310*j+59*k)
g = gcd(t,t2)
t /= g
t2 /= g
s = int(t**.5)
if s*s == t:
s2 = int(t2**.5)
if (s2*s2 == t2 and s*41 > 59*s2 and s*i <= 37760 and s*j <= 11328 and s*k <= 339840):
print i,j,k,s,s2
#
#
#
from Functions import gcd
def A(n):
curRepUnit = 1;
k = 0;
ctr=1
while (curRepUnit%n != 0):
curRepUnit = pow(10,ctr)/9
k+=1
ctr+=1
return k
##########################################
for n in xrange(1000020,1000200):
if gcd(n,10)!=1:continue
print n, A(n)
#
# Euler 73
#
from Functions import gcd
from math import ceil
from time import time
st=time()
ctr = -2
d={}
# mini = 1./3.
# maxi = .5
for i in xrange(1,12001):
x=int(ceil(i/3.))
#print x ,i
for j in xrange(x,int(i/2)+1 ):
#val = (1.0*j)/i
#print x
if gcd(i,j) == 1:
# if not( val>mini and val < maxi):
# print i,j
# continue
if i*12000+j not in d:
ctr +=1
d[i*12000+j]=1
print "total is ", ctr
print "time elapsed is ", time()-st
if gcd(i, j) != 1: continue
s = i + j
#print "M:",M,i,j,s
if s >= M:
print "M:", M, i, j, s
y = 1.0 / (i * j)
tot += y
ctr += 1
return tot, ctr
ctr = 0
d = {}
summ = 0
for i in xrange(2, 10001):
for j in xrange(i + 1, 10001):
if gcd(i, j) == 1:
ctr += 1
#print i,j
# d[(i,j)]=(1.0/(i*j))
print
#print d
print "Grand Total is ", summ
print "time elapsed is ", time() - st
# print
# print R(1000)