def nb2(lim): # n = p^3*q with n <= lim
s = 0
for i in range(len(l_p)): # q < p
p = l_p[i]
if p**3 > lim // 2:
break
s += min(binary_search_inf(l_p[:i + 1], lim / (p**3)) + 1, i)
for i in range(len(l_p)): # q > p
q = l_p[i]
b = lim / q
if b < 8:
break
s += min(binary_search_inf(l_p[:i + 1], b**(1 / 3)) + 1, i)
return s
def problem_49():
l4 = numpy_sieve(10**4)
i0 = binary_search_inf(l4, 1000)
l4 = l4[i0 + 1:]
def permut_ok(n):
m2 = str(n)
m = list(m2)
m.sort()
for i in range(1, 3):
n2 = n + i * 3330
if not binary_search(l4, n2)[0]:
return False, ""
n2 = str(n2)
nn2 = list(n2)
nn2.sort()
if nn2 != m:
return False, ""
m2 += n2
return True, m2
for p in l4:
if p != 1487:
ok, s = permut_ok(p)
if ok:
print(s) # sol = 296962999629
break
def nb1(lim): # n = p^7 with n <= lim
a = int(lim**(1 / 7))
if (a + 1)**7 < lim:
a += 1
if a < 2:
return 0
return binary_search_inf(l_p, a) + 1
def nb3(lim):
s = 0
for p in l_p: # p > q > r
for i in range(len(l_p)):
q = l_p[i]
if q >= p:
break
if q * p * 2 > lim:
break
s += min(binary_search_inf(l_p[:i + 1], lim / (p * q)) + 1, i)
return s
def problem_51():
l_7 = numpy_sieve(10**7)
i_start = binary_search_inf(l_7, 10**5)
def make_pattern(n):
m = str(n)
for digit in m:
if m.count(digit) == 3:
yield [
5 - i for i in range(len(m) - 1, -1, -1) if m[i] == digit
]
def apply_pattern(n, pat):
n2 = 0
k = 0
while n != 0:
n2 += (n % 10) * 10**k if k not in pat else 0
n //= 10
k += 1
return n2
def add(n, pat):
for i in pat:
n += 10**i
return n
viewed = set()
for p in l_7[i_start:]:
for pattern in make_pattern(p):
p2 = apply_pattern(p, pattern)
if p2 in viewed:
continue
viewed.add(p2)
nb = 0
for d in range(10):
if (5 not in pattern or d != 0) and binary_search(l_7, p2)[0]:
nb += 1
if nb + 9 - d < 8:
break
p2 = add(p2, pattern)
if nb < 8:
continue
print(p) # sol = 121313
return
elif n % 4 == 3:
k = n
p = lpd(k)
if p == 0 or p == k:
return True
return ok_aux(p, k)
else:
return False
def number(n):
nb = 0
for i in range(n):
if ok(i):
nb += 1
return nb
print("Problem 136: ", number(10**6*50))
# the 'good' n's are n == 4P or 16P or P where P prime and n == 4*1 and n == 16*1
# so :
maxi = 10**6*50
l_pp = numpy_sieve(maxi)
tot = 0
for a in l_pp:
if a % 4 == 3:
tot += 1
tot += binary_search_inf(l_pp, maxi//4) + 1
tot += binary_search_inf(l_pp, maxi//16) + 1
print(tot)
def ok(p):
pp = str(p)
for c in pp:
if c == '0':
return False
if pp.count(c) > 1:
return False
return True
l8bis = [prime for prime in l_8 if ok(prime)]
primes = [[]]
i = 0
k0 = 1
while k0 <= 8:
j = binary_search_inf(l8bis, 10**k0)
primes.append(l8bis[i:j + 1])
i = j + 1
k0 += 1
def _in(p):
p = str(p)
ll = list(p)
ll.sort()
return "".join(ll)
dict_in = [{} for _2 in range(10)]
for prime in l8bis:
nb_d = len(str(prime))
for i9 in range(i8 + 1, len_lp):
p9 = lp[i9]
if p2 * p1 * p3 * p4 * p5 * p6 * p7 * p8 * p9 > l:
break
lp2.append(p2 * p1 * p3 * p4 * p5 * p6 *
p7 * p8 * p9)
lp2.sort()
l2 = []
am = log(l) / log(2)
for a in range(int(am) + 1):
if a >= 32:
break
bm = (log(l) - a * log(2)) / log(3)
for b in range(int(bm) + 1):
cm = (log(l) - a * log(2) - b * log(3)) / log(5)
for c in range(int(cm) + 1):
n = 2**a * 3**b * 5**c
if n > l:
break
l2.append(n)
l2.sort()
s = sum(l2) % 2**32
for p in lp2:
k = binary_search_inf(l2, 10**12 / p)
s += p * sum(l2[:k + 1])
s %= 2**32
print(s) # 939087315