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):
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 cntpowers(x):
z = RetFact(x)
S = set(z)
summ = 0
for SS in S:
summ += z.count(SS)
return summ
def cntpowers(x):
z = RetFact(x)
S = set(z)
summ = 0
for SS in S:
summ += z.count(SS)
return summ
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 phi(n):
v = n
f = list(set(RetFact(n)))
for z in f:
v *= (z - 1)
v /= z
return v
def phi(n):
f = list(set(RetFact(n)))
num, div = 1, 1
for x in f:
num *= x - 1
div *= x
return n * num / div
def IsPhiStrong4(sa, phi):
flist = RetFact(phi)
for f1 in flist:
if phi % (f1 * f1) > 0: return False
#print "flist",sa,phi,flist
return True
def IsSAStrong(sa, factlist):
powlist = list(set(RetFact(sa)))
#print "!!",sa,powlist
for pp in powlist:
psquare = pp**2
if not (sa % psquare == 0):
#print "False!"
return False
return True
def divisors(n):
factors = RetFact(n)
factors.append(1)
fact = []
l = len(factors)
for z in xrange(2, l + 1):
y = combinations(factors, z)
for yy in y:
yyy = list(yy)
#print yyy
s1 = 1
for yyyy in yyy:
s1 *= yyyy
fact.append(s1)
fact.append(1)
fact = sorted(list(set(fact)))
#print fact
return fact
def A0(n):
for k in xrange(2, 17):
x = repunit(k)
if miller_rabin(x): continue
f = RetFact(x)
if n in f:
return k
return 0
def divisors(n):
factors = RetFact(n)
factors.append(1)
fact=[]
l = len(factors)
for z in xrange(2,l+1):
y = combinations(factors,z)
for yy in y:
yyy = list(yy)
#print yyy
s1 = 1
for yyyy in yyy:
s1*=yyyy
fact.append(s1)
fact.append(1)
fact = sorted(list(set(fact)))
#print fact
return fact
def Phi(n):
F = set(RetFact(n))
phi = n
num = 1
den = 1
for f in F:
num *= (f - 1)
den *= f
return n * num / den
def lcm(lst):
s = set()
for l in lst:
F = RetFact(l)
for f in F:
s.add(f)
lcm=1
print s
for i in s:
lcm *=i
return lcm
def Phi(Achnum, factlist):
# powlist = "["+factlist.rstrip(",")+"]"
# powlist = list(set(map(int,powlist.strip('[]').split(','))))
Phi = Achnum
powlist = list(set(RetFact(Phi)))
for i in powlist:
Phi = Phi * (int(i) - 1)
Phi = Phi / int(i)
return Phi
def tf(f):
t = list(f)
for f1 in f:
zz = RetFact(f1 - 1)
#print "zz",zz
for zzz in zz:
#print "zzz",zzz
if zzz not in f:
t.append(zzz)
t = sorted(list(set(t)))
return t
def PhiPrimes(n):
totfacts = []
n1 = n - 1
totfacts = RetFact(n1)
allfactsfound = False
while allfactsfound == False:
newfactfound = False
for m in totfacts:
o = RetFact(m - 1)
for p in o:
if p < 2: continue
if not p in totfacts:
totfacts.append(p)
newfactfound = True
if newfactfound == False: allfactsfound = True
return list(set(totfacts))
def PhiPrimes(n):
totfacts=[]
n1 = n-1
totfacts = RetFact(n1)
allfactsfound = False
while allfactsfound == False:
newfactfound = False
for m in totfacts:
o = RetFact(m-1)
for p in o:
if p<2:continue
if not p in totfacts:
totfacts.append(p)
newfactfound = True
if newfactfound == False: allfactsfound = True
return list(set(totfacts))
def divisorGen(n):
factors = list(RetFact(n))
nfactors = len(factors)
f = [0] * nfactors
while True:
yield reduce(lambda x, y: x*y, [factors[x]**f[x] for x in range(nfactors)], 1)
i = 0
while True:
f[i] += 1
if f[i] <= factors[i]:
break
f[i] = 0
i += 1
if i >= nfactors:
return
def Phi(Achnum, factlist):
#powlist = "["+factlist.rstrip(",")+"]"
#powlist = list(set(map(int,powlist.strip('[]').split(','))))
powlist = sorted(list(set(RetFact(Achnum))))
#if Achnum ==108:print "%%%%",powlist
Phi = Achnum
x1 = 1
x2 = 1
for i in powlist:
x1 = x1 * (int(i) - 1)
x2 = x2 * int(i)
Phi = (Phi * x1) / x2
return Phi
def IsPhiStrong3(sa, phi, factlist):
#powlist = "["+factlist.rstrip(",")+"]"
#powlist = list(set(map(int,powlist.strip('[]').split(','))))
#powlist = sorted(list(set(factlist)))
powlist = list(set(RetFact(phi)))
phi2 = phi
#print "sa,phi:",sa,phi2,powlist
for i in powlist:
#if phi==200:print "#",phi,i,phi2 % (i * i)
if not (phi2 % (i * i) == 0): return False
#if phi==200:print phi2%(i-1),phi2 % ((i-1) * (i-1))
if phi2 % (i - 1) == 0:
z = i - 1
if z % 4 == 0: z = z / 2
if phi2 % (z * z) > 0: return False
return True
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
Sum = sum(x)
maxlist=[]
#print tmp
#tmp=list(x)
for l in product(*W):
#l=sorted(l,reverse = False)
tmp=list(x)
#if GCD(l)>1:continue
#s = sum(l)
#print l, s
tsum = 0
for j in l:
#print j
if j in tmp:continue
f = sorted(list(set(RetFact(j))))
#print "F",f
if len(f)==1:
old = notcoprime(f[0],tmp)
#print "Old",old
if j > old:
tsum+=j-old;tmp.append(j);tmp.remove(old)
continue
#print "j",j,f
f1 = f[0];f2 = f[1]
if j/f1!=f2:
f1 = j/f2
elif j/f2!=f1:
f2 = j/f1
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
# for x in d:
# z = RetFact(x)
# two = z.count(2)
# zz = x/(2**two)
# zzz = 3**(two+1)*zz
# if zzz>(120120/4):
# print x,two, zz,z
#
#
#
#
#
from Functions import RetFact
d = 12000.
f = RetFact(d)
ctr = 0
for i in xrange(4001, 6000):
#if not (d%i == 0): ctr+=1
a = RetFact(i)
if bool(set(a) & set(f)) == False:
print i, ":", a
ctr += 1
print ctr
u = int(1999. / 3.)
v = int(1999. / 2.)
w = int(1999. / 5.)
x = int(1999. / 6.)
y = int(1999. / 10.)
z = int(1999. / 15.)
z1 = int(1999. / 30.)
tot = u + v + w - x - y - z #-z1
print tot, 1999 - tot
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
factlist = str(i[0]) + "," + str(j[0]) + "," + str(k[0])
phi = Phi(sa, factlist)
if perfect(phi): continue
#if sa < lim and IsPhiStrong3(sa,phi,factlist):
if sa < lim and IsPhiStrong4(sa, phi):
#print i,j,k ,sa, factlist, phi
cnt += 1
SAset.append(sa)
x = list(set(SAset))
x.sort()
for b in x:
u = RetFact(b)
uu = pfset(u)
print b, uu #RetFact(b)
#print "Number of numbers is", x
print
print
#print type(x), len(x), type(x[1])
print "number of primes is", len(x)
def phi(n):
f = list(set(RetFact(n)))
num,div = 1,1
for x in f:
num *= x-1
div *= x
return n*num/div
N=1000000000L
ctr=0
for i in xrange(N-1000000,N):
sq = phi(i)*i
cbr = int(round(sq**(1./3.),0))
if cbr**3 ==sq:
print "Hit!",i,i*i,sq,cbr,RetFact(i)
ctr+=i
print "total for ",N,"is",ctr
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
#
# Euler 204
#
from Functions import RetFact, IsPrime, primes
print "counting primses..."
#pns = primes(10**9)
print "primes calculated..."
summ = 99
for i in xrange(100, 10**9 + 1):
#if (i in pns):continue
zz = RetFact(i)
if max(zz) > 100: continue
#print i,zz
summ += 1
print
print "Hamming Number type 100 <10^9 is", summ + 1
#
#
#
# x + y = xy/n
#
# 1/x = 1/n - 1/y
#
#
#
from Functions import RetFact, IsPrime
n = 1260 #500000000
maxx = 0
for n in xrange(10**8, 10**9): #15000):
if IsPrime(n):continue
x = RetFact(n**2)
s = set(x)
facts = []
for f in s:
facts.append(x.count(f)+1)
m=1
for f in facts:
m*=f
m+=1
m/=2
if m>maxx:
maxx=m
print "!!!!!",n,":",m
else:
print n,":",m
if maxx>4000000:print "Over 4 Mil!",n,m
#
# 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
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
if ctr>0:
print
print p,q,ctr, ":", RetFact(q-1)
print ii,jj,vv
print imax,jmax
print "###############"
sctr+=ctr
Prev = Old1 + zz
#print New>Prev
if New > Prev:
makechange = True
if zprev + New > nprev + zz:
nprev = New
zprev = zz
Oprev = Old1
print vsing, zz, nprev
if vsing >= int(MX * .9):
print "grrr"
if vsing not in x: print "GRRRRRRR"
makechange = False
if makechange == True:
print z, Oprev
x.remove(Oprev)
x.remove(zprev)
x.append(nprev)
#x.append(1)
#print sorted(x)
#print MX,":",sorted(x),":",Sum(x)
print MX, ":", x, ":", Sum(x)
print "Process time is", time() - st
for i in x:
if not IsPrime(i): #print i # i%2==0:
print i, RetFact(i)
#
#
# Euler 124
#
# []
#
ls = []
from Functions import RetFact, RofPrimes
primes = RofPrimes(2, 100001)
for n in xrange(1, 100001):
if n in primes:
ls.append((n, n))
continue
x = list(set(RetFact(n)))
tot = 1
for y in x:
tot *= y
ls.append((tot, n))
ls = sorted(ls)
print ls[10000], ls[9999]
print "done"
lim = 100000 # 10000
# F = FactorSieve(100**3)
a, b, c = 0, 0, 0
cnt = 0
for a in xrange(2, lim, 3):
x = 8 * a * a * a + 15 * a * a + 6 * a - 1
if x % 27 == 0:
x /= 27 # x = b*b*c
y = RetFact(x)
y.append(1)
y = set(y)
# print "!",a,y
for z in y:
b = z
c = int(x / (b ** 2))
# print "!!",x,":",a,b,c
if x == b * b * c:
if a + b + c <= 1000: # 110000000:
cnt += 1
# print a,b,c
print "Number of Cardano Triplets <=", lim, "is", cnt
print "Process time is", time() - st
for x in xrange(2, len(d)):
f1 = combinations(d, x)
for f in f1:
prod = Prod(f)
if prod == i:
summ = sum(f)
k = len(f) + prod - summ
#print i,prod, summ, len(f), k
if k not in psnum:
psnum[k] = i
elif i < psnum[k]:
print "%"
psnum[k] = i
d = RetFact(i)
if len(set(d)) == 1:
prod = Prod(d)
summ = sum(d)
k = len(d) + prod - summ
#print i,prod, summ, len(d), k
if k not in psnum:
psnum[k] = i
elif i < psnum[k]:
#print "&"
psnum[k] = i
print
#print psnum
dupes = []
tot = 0
