def ffffv1v2(N):
"f*****g fast factorial factoring -- edit"
rt = int(math.sqrt(N))
prms = [True]*(N+1)
prms = sieve(N)
bits = [N//k for k in range(1, int(math.sqrt(N))+1)] #<-- actual buckets
#bits.reverse()
#print(bits)
#buckets = {}
d = {}
for k in range(len(bits)-1): #NB: only goes up to sqrt(N)-1
#print("(%d, %d]" % bounds)
for i in range(bits[k+1]+1, bits[k]+1):
if prms[i]:
d[i] = k+1
#buckets[k+1] = [i for i in if prms[i]]
#print(buckets)
def c(b):
"count the number of powers of b in N!"
m = int(math.log(N, b))
return sum(N//b**j for j in range(1, m+1))
for i in range(2, int(math.sqrt(N))+1):
if prms[i]:
d[i] = c(i)
#for k in buckets:
# for e in buckets[k]:
# d[e] = k
return d
def ffffv4(N):
S = sieve(N)
factorization = {}
#important f*****g line:
#also, magical f*****g line.
#k is a bucket index, (N//(k+1), N//k] is a bucket interval for bucket k. but many buckets are lamely empty,
#however these lines compute *only* the nonempty buckets
bits = [N//k for k in range(1, int(math.sqrt(N)))] #<-- actual buckets
bits += [g for g in range(bits[-1]-1, 0, -1)] #<-- single elements
bits.reverse()
#print(bits)
for i,k in enumerate(bits[:-1]):
Range = bits[-(i+1)-1]+1, bits[-(i+1)]+1
#print(i, k, Range)
#block = S[Range[0]:Range[1]]
#print("=>", block)
for i in range(*Range):
p = i #S[i] #[p for p in block if p > 0]:
if not S[p]: continue
#if k < math.sqrt(N):
# print(i,p)
factorization[p] = factorization.get(p, 0) + k
#print(prettyprint(factorization))
def c(b):
"count the number of powers of b in N!"
m = int(math.log(N, b))
return sum(N//b**j for j in range(2, m+1))
i = 0
for i in range(2, int(math.sqrt(N))+1):
if S[i]:
factorization[i] += c(i)
#prms = [i for i in range(2, int(math.sqrt(N))+1) if prms[i]]
#print(prettyprint(factorization))
return factorization