def fi(n):
v=primes(int(n**0.5)+1)
for i in range(3,n+1,2):
if factornum(i,v)==3:
x=isresult(i,v)
if x:
return x
def fi():
v=primes(100)
print(v)
s=1
i=0
while s<maxlen:
s=s*v[i]
i=i+1
print(s)
s=s//v[i-1]
return s
def f():
ps=primes(1000100)
i=2
re=0
while ps[i]<=1000000:
p=ps[i]
t=10**(int(log(p,10))+1)
m=ps[i+1]
j=g(t,m-p,m)%m
## print(t*j+p)
re+=t*j+p
i+=1
return re
def f(n):
L=list(factorial(n))
S=set()
ps=primes(n)
def C(m,k):
return L[m]//(L[k]*L[m-k])
def notsquarefree(x):
return all(x%(i*i)!=0 for i in ps)
for i in range(n):
for j in range(i//2+1):
t=C(i,j)
print(i,j,t)
if notsquarefree(t):
S.add(t)
## print(S)
return sum(S)
def p70():
x=primes(10000)
t,s=0,3
xa=[]
for i in x:
for j in x:
if j<i:
continue
if permutation(i*j,(i-1)*(j-1)) and i*j/((i-1)*(j-1))<s:
print(i,j)
t,s=i*j,i*j/((i-1)*(j-1))
xa.append(i*j)
if i*j>10**7:
break
if i*i>10**7:
break
return t
def p84(n):
result=set()
ps=primes(int(n**0.5)+1)
for c in ps:
x=c**4
if x>=n:
break
for b in ps:
y=x+b**3
if y>=n:
break
for a in ps:
z=y+a**2
if z>=n:
break
result.add(z)
return len(result)
from functools import reduce
from Crazy import primes
ps=primes(1000)
def products(l):
return reduce(lambda x,y:x*y,l)
def sod(divs):
return products([(i**(divs[i]+1)-1)//(i-1) for i in divs])
def g(n):
divs={}
m=n
for i in ps:
if i*i>n:
break
while n%i==0:
n=n//i
if i in divs:
divs[i]+=1
else:
divs[i]=1
if n not in divs and n!=1:
divs[n]=1
return sod(divs)-m
def f1(n):
t=1
for i in range(2,n+1):
if i%10000==0:
print(i)
m=g(i)
j=1
L=[]
while m>i and m not in L and m<=n:
##The primes 3, 7, 109, and 673, are quite remarkable. By taking any two primes
##and concatenating them in any order the result will always be prime. For example
##, taking 7 and 109, both 7109 and 1097 are prime. The sum of these four primes,
##792, represents the lowest sum for a set of four primes with this property.
##Find the lowest sum for a set of five primes for which any two primes
##concatenate to produce another prime.
from Crazy import primes
from Crazy import isprime
filen=8
ps=primes(10**4)
def conisprime(n,m):
return isprime(int(str(n)+str(m))) and isprime(int(str(m)+str(n)))
def fi():
re={}
for i in range(1,len(ps)):
if len(str(ps[i]))<=4:
for j in range(i):
if len(str(ps[j]))<=filen-len(str(ps[i])) and conisprime(ps[i],ps[j]):
if ps[j] in re:
re[ps[j]].append(ps[i])
else:
re[ps[j]]=[ps[i]]
for i in re:
for j in range(len(re[i])):
if len(re[i])>=4 and re[i][j] in re:
for k in range(j+1,len(re[i])):
if re[i][k] in re[re[i][j]] and re[i][k] in re:
for l in range(k+1,len(re[i])):
if re[i][l] in re[re[i][j]] and re[i][l] in re[re[i][k]] and re[i][l] in re:
for m in range(l+1,len(re[i])):
from Crazy import primes
from fractions import Fraction
X=500
P=primes(500)
croaks="PPPPNNPPPNPPNPN"
##croaks="NN"
def p329():
Prob=[]
L={"N":[],"P":[]}
for i in range(1,X+1):
if i in P:
L["N"].append(Fraction(1,3))
L["P"].append(Fraction(2,3))
else:
L["N"].append(Fraction(2,3))
L["P"].append(Fraction(1,3))
## print(L)
Re=L[croaks[-1]]
for c in reversed(croaks[:-1]):
## print(croaks[:-1],croaks[-1])
T=L[c]
Temp=[]
for i in range(0,X):
x=T[i]
if i==0:
x=x*Re[1]
elif i==X-1:
x=x*Re[i-1]
else:
x=x*Fraction(1,2)*(Re[i-1]+Re[i+1])
Temp.append(x)
##It is possible to write ten as the sum of primes in exactly five different
##ways:
##
##7 + 3
##5 + 5
##5 + 3 + 2
##3 + 3 + 2 + 2
##2 + 2 + 2 + 2 + 2
##
##What is the first value which can be written as the sum of primes in over five
##thousand different ways?
MAX = 100
from Crazy import isprime, primes
ps = primes(MAX)
def p77(num):
def cachnum(n, k):
return k * (MAX + 1) + n
Cached = {}
for n in range(MAX):
for k in range(MAX):
x = cachnum(n, k)
if n == 0:
Cached[x] = 1
elif n < k:
Cached[x] = 0
elif n == k:
Cached[x] = 1
def f1(n):
ps = primes(3 * 10 ** 5)
for i, j in enumerate(ps, 1):
if i & 1 == 1:
if ((2 * i) % j) * j > n:
return i, j
