import proj_euler as pe
import numpy as np
mod = 10**9+7
max_n = 10**4+10
fact, ifact = pe.mod_fact_cache(mod, max_n = max_n)
def comb(n,k):
if k > n: return 0
if n < mod:
return (((int(fact[n]) * int(ifact[k])) % mod )* int(ifact[n-k])) % mod
n_digits, k_digits = pe.base_repr(n, mod), pe.base_repr(k, mod)
res = 1
for top, bot in zip(n_digits, k_digits):
res = (res * C(top, bot)) % mod
return res
def double_power_set(n):
#for prime mods, the totient is p-1
return pow(2, pow(2, n, mod-1) -1 ,mod)
def C(n,k):
#build cache tables
cache = np.zeros((n+1, k+1), dtype=int)
exact = np.zeros(n+1, dtype=int)
for j in xrange(1, n+1):
if j%100 == 0:
print "on "+str(j)
#all possible graphs
cache[j,1] = cache[j-1,1] + double_power_set(j) - double_power_set(j-1)
import proj_euler as pe
import numpy as np
#n = 5*10**4
mod = 1000000123
fact, ifact = pe.mod_fact_cache(mod, max_n = 6*10**4+2)
def P(k, r, n):
if k == n: return 1
q, rem = divmod(n-1,k)
rem += 1
dividers = q
divs = np.zeros(dividers+2,np.int64)
#number of pieces available
divs[0] = 1
divs[1] = pow(int(ifact[k]), r, mod)
pows = np.array([(-1) ** (j%2) * pow(int(ifact[j*k]), r, mod) for j in xrange(dividers+1)])
for i in xrange(2, dividers+1):
temp = np.remainder(pows[1:i+1] * divs[i-1::-1], mod)
divs[i] -= np.sum(temp)
divs[i] %= mod
for j in xrange(1, len(divs)):
# print divs[-j-1], j
divs[-1] -= (-1) ** (j%2) * pow(int(ifact[(j-1)*k+rem]), r, mod) * divs[-j-1]
divs[-1] %= mod
return (divs[-1] * pow(int(fact[n]), r, mod)) % mod
def Q(n):
tot = 0
for k in xrange(1,n+1):
if k < 10:
print "On %d" %k