#So the sum of the terms in the prime factorisation of 10C3 is 14.
#Find the sum of the terms in the prime factorisation of 20000000C15000000.
#Answer:
#7526965179680
from time import time
t = time()
#from mathplus import get_primes_by_sieve
from gen_primes import get_primes
N, M = 20000000, 15000000
M = min(M, N - M)
#primes, sieves = get_primes_by_sieve(N)
primes = get_primes(50000000)
#print time()-t
def factfact(n):
#ret = {}
ret = 0
for p in primes:
if p > n: break
k, m = 0, n
while m >= p:
m //= p
k += m
#ret[p] = k
ret += p * k
return ret
#How many composite integers, n < 108, have precisely two, not necessarily distinct, prime factors?
#Answer:
#17427258
from time import time
t = time()
from mathplus import isqrt, get_primes_by_sieve
from gen_primes import get_primes
#M = 30
M = 10**8
sqrtM = isqrt(M)
primes = get_primes(M // 2, odd_only=False)
#print time()-t
sieves = [0] * (sqrtM + 1)
pnext = 2
pnext_index = 0
p = 0
for i in range(sqrtM + 1):
if i == pnext:
pnext_index += 1
pnext = primes[pnext_index]
p += 1
sieves[i] = p
#print time()-t
ss = sieves[sqrtM]
s = ss * (ss + 1) // 2
s += sum(sieves[M // p] for p in primes[ss:])
#There are ten composites below thirty containing precisely two, not necessarily distinct, prime factors: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26.
#How many composite integers, n < 108, have precisely two, not necessarily distinct, prime factors?
#Answer:
#17427258
from time import time; t=time()
from mathplus import isqrt, get_primes_by_sieve
from gen_primes import get_primes
#M = 30
M = 10**8
sqrtM = isqrt(M)
primes = get_primes(M//2, odd_only=False)
#print time()-t
sieves = [0]*(sqrtM+1)
pnext = 2
pnext_index = 0
p = 0
for i in range(sqrtM+1):
if i == pnext:
pnext_index += 1
pnext = primes[pnext_index]
p += 1
sieves[i] = p
#print time()-t
ss = sieves[sqrtM]
s = ss*(ss+1)//2
s += sum(sieves[M//p] for p in primes[ss:])