def IsPractical(n):
D = divisors(n)
D.remove(n)
l = len(D)
slist = [1, n]
for i in xrange(1, l + 1):
C = combinations(D, i)
for c in C:
s = sum(c)
#print i, c, s
if s <= n and s not in slist:
slist.append(s)
#elif s>n+6:break
if len(slist) == n:
print "*", n
return True
else:
print "##"
return False
def IsPractical(n):
if n % 2 != 0: return False
#D = divisors(n)
#if sum(D) == n*2: return True
lg = log(n) / log(2)
if lg == int(lg): return True
F = RetFact(n)
S = sorted(list(set(F)))
for k in range(1, len(S)):
a = S[k]
f = S[:k]
v = 1
for fn in f:
v *= fn**F.count(fn)
w = sum(divisors(v)) + 1
if a > w:
print a, w, n, S
return False
print "Yes", n
return True
def IsPractical(n):
D = divisors(n)
D.remove(n)
l = len(D)
slist = [1,n]
for i in xrange(1,l+1):
C = combinations(D,i)
for c in C:
s = sum(c)
#print i, c, s
if s<=n and s not in slist:
slist.append(s)
#elif s>n+6:break
if len(slist) == n:
print "*",n
return True
else:
print "##"
return False
def find_chain(n):
ap = False
n0 = n
chain = [n0]
strt = n0
while 1:
d = divisors(n0)
#if n==276:print d
nw = sum(d) - n0
#print nw
if nw == n and len(chain) == 1:
#print "!!!!! Perfect # found"
break
if nw == n:
#print "!!!!! Amicable chain found"
ap = True
break
elif nw == 1:
chain.append(1)
break
elif nw in chain:
break
elif n0 > 999999:
break
else:
chain.append(nw)
n0 = nw
return chain, ap
def IsPractical(n):
if n%2 != 0: return False
#D = divisors(n)
#if sum(D) == n*2: return True
lg = log(n)/log(2)
if lg == int(lg): return True
F = RetFact(n)
S = sorted(list(set(F)))
for k in range(1,len(S)):
a = S[k]
f = S[:k]
v = 1
for fn in f:
v*= fn**F.count(fn)
w = sum(divisors(v)) + 1
if a > w:
print a, w,n,S
return False
print "Yes",n
return True
def find_chain(n):
ap = False
n0 = n
chain = [n0]
strt = n0
while 1:
d = divisors(n0)
#if n==276:print d
nw = sum(d) - n0
#print nw
if nw == n and len(chain)==1:
#print "!!!!! Perfect # found"
break
if nw == n:
#print "!!!!! Amicable chain found"
ap = True
break
elif nw ==1:
chain.append(1)
break
elif nw in chain:
break
elif n0>999999:
break
else:
chain.append(nw)
n0 = nw
return chain,ap
def O2(n):
tot = 0
D = divisors(n)
#print n,D
for d in D:
tot += d * d
return tot
def O2(l):
tot = 0
D = divisors(n)
#print n,D
for d in D:
tot += d*d
return tot
def f(n):
if not IsPrime(1 + n): return False
if not IsPrime(2 + n / 2): return False
D = divisors(n)
D = D[:len(D) / 2]
#l = len(D)
for d in D:
if not IsPrime(d + n / d): return False
return True
def f(n):
if not IsPrime(1 + n):return False
if not IsPrime(2 + n/2):return False
D = divisors(n)
D = D[:len(D)/2]
#l = len(D)
for d in D:
if not IsPrime(d + n/d):return False
return True
def IsSF(d):
v = 0
D1 = divisors(d)
D1.remove(1)
for d1 in D1:
if d1**.5 == int(d1**.5):
return False
return True
def IsPractical(n):
D = divisors(n)
for k in xrange(2,n):
if k in D:continue
f = bisect(D,k)
if not InD(k,D[:f]):
#print n,k
return False
#print "Yes"
return True
def IsPractical(n):
D = divisors(n)
for k in xrange(2, n):
if k in D: continue
f = bisect(D, k)
if not InD(k, D[:f]):
#print n,k
return False
#print "Yes"
return True
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
result = 1
for fs in facts:
result *= fs
return result
#psnum = {}
d={}
stor=[]
for i in range(2,1001):
if IsPrime(i):continue
v = divisors(i)
f0 = RetFact(i)
f=f0
v.pop(-1)
v.pop(0)
for x in v:
if x not in f:
f += [x]
f = sorted(f)
#print f
k=0
#print "*",len(f),f
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:
summm +=1
print "*",x,RetFact(x)
#print "*",x,RetFact(x)
#if D==X: print
print summm
print "time elapsed", time()-st
#
#
# Euler Problem 283
#
#
from Functions3 import GCD,gcd
from math import sqrt
from Functions import divisors
def ta(a,b,c):
A = (((a+b+c)*(a+b-c)*(a+c-b)*(b+c-a))**.5)/4
return A
for m in xrange(1,10):
f = 8*m**2
fd = divisors(f)
print m,f,fd
ul = 2*m*2**.5
for d1 in fd:
if d1 <=ul:
d2 = f/d1
a = d1 + 4*m
b = d2 +4 *m
c = d1+d2+4*m
A = ta(a,b,c)
P = a+b+c
print " :",d1, d2,":",a,b,c,":",A,P,A/float(P)
'''
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
# summ+=2
# continue
# summ +=sl
def canAdd(s,y):
#canAdd=True
for z in s:
if y>z:
if y/z==(1.0 * y)/z:
return False
else:
if z/y==(1.0 * z)/y:
return False
return True
d = divisors(120120)
x=[]
s=set(x)
s2=set(x)
for i in xrange(1,128):
s = set([d[i]])
for j in xrange(i+1,128):
#if d[j]%5==0:continue
if canAdd(s,d[j]):
s=s.union(set([d[j]]))
for j in xrange(2,128):
#if d[j]%5==0:continue
if (d[j] not in s) and canAdd(s,d[j]):
s=s.union(set([d[j]]))
#
# Euler 380
#
#
from Functions import RetFact,divisors
ctr = 0
for b in xrange(1,333334):
ctr+=1
b2=b**2
f = divisors(b2)
#f.remove(1)
f.remove(b2)
l = len(f)
for i in xrange(l-1,1,-1):
c=f[i]
a = b2/c
if a<=b<=c and a+c>b and max(a,c)-min(a,c)<b:
print b, a, c
ctr+=1
else: continue
print ctr
def ta(a, b, c):
A = (((a + b + c) * (a + b - c) * (a + c - b) * (b + c - a)) ** 0.5) / 4
return A
summ = 0
#
# Using Heron's formula from Lubomir Markov's extension of Goehl's Method
#
M = []
for m in xrange(1, 1001):
print "m:", m,
psum = 0
# u = 2*m
fd0 = divisors(2 * m) # Factoring u = 2m
for u in fd0:
ul = u * 3 ** 0.5 # choosing factors up to u*sqrt(3)
mxul = int(ul) + 1
for v in xrange(1, mxul):
if gcd(u, v) != 1:
continue # choosing v coprime to u
f = 4 * m * m * (u * u + v * v) # = delta1 * delta2
fd = divisors(f) # obtaining factors for d1,d2
ul2 = 2 * m * ((u * u + v * v) ** 0.5) # obtaining sqrt to avoid repetitions
# print "!",m,u,v # showing m,u,v for curiosity's sake
for d1 in fd:
if ((2 * m) * (v ** 2 - u ** 2) / u) <= d1 <= ul2: # choosing range of delta1 to ensure A=mP
d2 = f / d1
a = (d1 + 2 * m * u) / v + (2 * m * v) / u
b = (d2 + 2 * m * u) / v + (2 * m * v) / u
# # Euler Problem 359 # # from Functions import RetFact, divisors facts = RetFact(71328803586048) rooms = divisors(71328803586048) print len(rooms) # 1 1,1 # 2 2,1 # 3 1,2 = 4 # 4 3,1 # 5 4,1 # 6 3,2 = 9 # 7 2,2 = 9 # 8 1,3 = 9 # 9 5,1 # 10 6,1 # 11 5,2 = 16 # 12 4,2 = 16 # 13 3,3 = 16 # 14 2,3 = 16 # 15 1,4 = 16 # 16 7,1 # 17 8,1
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 Problem 283
#
#
from Functions3 import GCD, gcd
from math import sqrt
from Functions import divisors
def ta(a, b, c):
A = (((a + b + c) * (a + b - c) * (a + c - b) * (b + c - a))**.5) / 4
return A
for m in xrange(1, 10):
f = 8 * m**2
fd = divisors(f)
print m, f, fd
ul = 2 * m * 2**.5
for d1 in fd:
if d1 <= ul:
d2 = f / d1
a = d1 + 4 * m
b = d2 + 4 * m
c = d1 + d2 + 4 * m
A = ta(a, b, c)
P = a + b + c
print " :", d1, d2, ":", a, b, c, ":", A, P, A / float(P)
'''
summ = 0
ctr,ctr1 = 0,0
for m in xrange(2, 4000):
# summ+=2
# continue
# summ +=sl
def canAdd(s,y):
#canAdd=True
for z in s:
if y>z:
if y/z==(1.0 * y)/z:
return False
else:
if z/y==(1.0 * z)/y:
return False
return True
d = divisors(120120)
x=[]
s=set(x)
s2=set(x)
for x in d:
z = RetFact(x)
S = set(z)
summ = 0
for SS in S:
summ += z.count(SS)
if summ == 4:
print x
from Functions import RetFact, divisors,IsPrime
summ = 6
for n in xrange(3, 6):
if IsPrime(n):
print n,n%4
if n%4==1 :
pass
else:
summ += (n+1)
else:
d = divisors(n)
print n,d
summ +=sum(d)
print
print summ
#
# 6 20 1 ,6, 3,
# 7 8 1 ,7
# 8 29
# 9 13
# 10 56
# 11 12
# 17
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
from Functions import RetFact, divisors, IsPrime
summ = 6
for n in xrange(3, 6):
if IsPrime(n):
print n, n % 4
if n % 4 == 1:
pass
else:
summ += (n + 1)
else:
d = divisors(n)
print n, d
summ += sum(d)
print
print summ
#
# 6 20 1 ,6, 3,
# 7 8 1 ,7
# 8 29
# 9 13
# 10 56
# 11 12
# 17
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
return A
summ = 0
#
# Using Heron's formula from Lubomir Markov's extension of Goehl's Method
#
M = []
from time import time
st = time()
for m in xrange(1, 1001):
#print "m:",m,
psum = 0
#u = 2*m
fd0 = divisors(2 * m) # Factoring u = 2m
for u in fd0:
ul = u * 3**.5 # choosing factors up to u*sqrt(3)
mxul = int(ul) + 1
for v in xrange(1, mxul):
if gcd(u, v) != 1: continue # choosing v coprime to u
f = 4 * m * m * (u * u + v * v) # = delta1 * delta2
fd = divisors(f) # obtaining factors for d1,d2
ul2 = int(
2 * m *
((u * u + v * v)**.5)) # obtaining sqrt to avoid repetitions
#print "!",m,u,v # showing m,u,v for curiosity's sake
for d1 in fd:
if ((2 * m) * (v**2 - u**2) / u
) <= d1 <= ul2: #choosing range of delta1 to ensure A=mP
if (d1 + 2 * m * u) % v: continue
from Functions import divisors,IsPrime
def IsSF(d):
v = 0
D1 = divisors(d)
D1.remove(1)
for d1 in D1:
if d1**.5 == int(d1**.5):
return False
return True
summ = 0
for k in xrange(2,101):
cnt = 0
D = divisors(k)
D.remove(1)
for d in D:
if IsSF(d):
cnt +=1
print "!",k,d,cnt
summ += cnt
print k, cnt
print summ
