mp.append(sum(arr))
# print(mp)
for i in range(len(mp)):
# print(i, mp[i - 1], mp[i])
for j in range(mp[i - 1], mp[i]):
out[j] = i
# out[mp[len(mp) - 1]] = len(mp) - 1
return tuple([0] + out)
pm = {2: p_to_m(2), 3: p_to_m(3), 5: p_to_m(5), 7: p_to_m(7)}
for x in pm:
print(x, pm[x], len(pm[x]))
M = 10**8
pp = mylib.primes_from_file(M)
f_cashed = dict()
def get_f(dd):
if dd in f_cashed:
return f_cashed[dd]
# print(' ', dd)
d = dd[0]
p = len(dd)
if d in pm:
p = pm[d][p]
f_cashed[dd] = d * p
return f_cashed[dd]
If N is admissible, the smallest integer M > 1 such that N+M is prime, will be called the pseudo-Fortunate number for N.
For example, N=630 is admissible since it is even and its distinct prime factors are the consecutive primes 2,3,5 and 7.
The next prime number after 631 is 641; hence, the pseudo-Fortunate number for 630 is M=11.
It can also be seen that the pseudo-Fortunate number for 16 is 3.
Find the sum of all distinct pseudo-Fortunate numbers for admissible numbers N less than 10**9.
"""
import math
import mylib
top = 10 ** 9
pp_100 = sorted(mylib.primes_from_file(100))
bb = []
m = 1
for x in pp_100:
m *= x
if m >= top:
m //= x
break
bb.append(x)
print(bb, m)
rrr = [[1]]
for b in bb:
mm = sorted(rrr[len(rrr) - 1])
p = math.floor(math.log(top, b))
nn = [b ** (x + 1) for x in range(p)]
import mylib
import math
L = 20
r = math.floor(L ** 0.5)
pp = sorted(x for x in mylib.primes_from_file(r) if x <= r)
print(pp)
for m in pp:
print(m)
import mylib, math
pp = sorted(mylib.primes_from_file(10 ** 6 + 1))
ends = dict((x, [x * y % 10 for y in range(10)]) for x in [1, 3, 7, 9])
# print(ends)
ends_map = dict()
for p in ends:
ends_map[p] = [0 for x in range(10)]
for i in range(len(ends[p])):
ends_map[p][ends[p][i]] = i
print(ends_map)
def divide(x, y):
x_str = str(x)
z = x * 1
pwr = len(x_str)
d = 0
e = y % 10
print('*' + str(x), '/', y, end=' ')
for c in range(pwr):
n = z % 10
m = ends_map[e][n]
d += m * 10 ** c
z = (z - m * y) // 10
print(' > ', n, m, m * y, end=' ')
print(' : ', d * y, '/', y, '=', d)
return d * y
mp.append(sum(arr))
# print(mp)
for i in range(len(mp)):
# print(i, mp[i - 1], mp[i])
for j in range(mp[i - 1], mp[i]):
out[j] = i
# out[mp[len(mp) - 1]] = len(mp) - 1
return tuple([0] + out)
pm = {2: p_to_m(2), 3: p_to_m(3), 5: p_to_m(5), 7: p_to_m(7)}
for x in pm:
print(x, pm[x], len(pm[x]))
M = 10 ** 8
pp = mylib.primes_from_file(M)
f_cashed = dict()
def get_f(dd):
if dd in f_cashed:
return f_cashed[dd]
# print(' ', dd)
d = dd[0]
p = len(dd)
if d in pm:
p = pm[d][p]
f_cashed[dd] = d * p
return f_cashed[dd]
import mylib, math
m = 500500507
t = 500500
pp = sorted(mylib.primes_from_file(10**7))[:t]
p_max = max(pp)
p_max_r = math.ceil(p_max**0.5)
print(len(pp), p_max, p_max_r, p_max_r**2)
rr = pp[:t]
qq = [q for q in pp if q <= p_max_r]
print(qq)
for q in qq:
x = q**2
while x < p_max:
print(q, x)
rr.append(x)
x = x**2
rr = sorted(rr)[:t]
print(max(rr))
s = 1
for i in range(len(rr)):
r = rr[i]
if not i % 10:
print(i, r)
s = (s * r) % m
print(s)
def init_primes(p):
mylib.known_primes = mylib.primes_from_file(10 ** math.ceil(p / 2))
print('primes:', '... ' + ', '.join([str(x) for x in (sorted(mylib.known_primes)[-10:])]))
import mylib
e = 4
ppp = mylib.primes_from_file(10 ** (2 * e))
pp = sorted(p for p in ppp if p < 10 ** e)
ss = [{}]
max_len = 0
for p in pp:
if p == 2:
continue
ss_add = []
for s in ss:
if not len(s):
ss_add.append({p})
continue
all_primes = True
for q in s:
pq = [str(p), str(q)]
if int("".join(pq)) not in ppp or int("".join(pq[::-1])) not in ppp:
all_primes = False
break
if all_primes:
s_new = s | {p}
if len(s_new) >= max_len:
max_len = len(s_new)
print(s_new)
ss_add.append(s_new)
ss += ss_add
print([(s, sum(s)) for s in ss if len(s) >= max_len])
import mylib, math
m = 500500507
t = 500500
pp = sorted(mylib.primes_from_file(10 ** 7))[:t]
p_max = max(pp)
p_max_r = math.ceil(p_max ** 0.5)
print(len(pp), p_max, p_max_r, p_max_r ** 2)
rr = pp[:t]
qq = [q for q in pp if q <= p_max_r]
print(qq)
for q in qq:
x = q ** 2
while x < p_max:
print(q, x)
rr.append(x)
x = x ** 2
rr = sorted(rr)[:t]
print(max(rr))
s = 1
for i in range(len(rr)):
r = rr[i]
if not i % 10:
print(i, r)
s = (s * r) % m
print(s)
"""
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?
"""
import mylib
pp = sorted(mylib.primes_from_file(100))
def find_ways(number, max_a=0):
ways = []
for a in pp:
if a > number:
break
if max_a and a > max_a:
break
if number == a:
ways.append([a])
continue
append_ways = find_ways(number - a, a)
for w in append_ways:
ways.append([a] + w)
return ways
def init_primes(p):
mylib.known_primes = mylib.primes_from_file(10**math.ceil(p / 2))
print(
'primes:', '... ' +
', '.join([str(x) for x in (sorted(mylib.known_primes)[-10:])]))
import mylib
import math
L = 20
r = math.floor(L**0.5)
pp = sorted(x for x in mylib.primes_from_file(r) if x <= r)
print(pp)
for m in pp:
print(m)
import mylib
e = 4
ppp = mylib.primes_from_file(10**(2 * e))
pp = sorted(p for p in ppp if p < 10**e)
ss = [{}]
max_len = 0
for p in pp:
if p == 2:
continue
ss_add = []
for s in ss:
if not len(s):
ss_add.append({p})
continue
all_primes = True
for q in s:
pq = [str(p), str(q)]
if int(''.join(pq)) not in ppp or int(''.join(
pq[::-1])) not in ppp:
all_primes = False
break
if all_primes:
s_new = s | {p}
if len(s_new) >= max_len:
max_len = len(s_new)
print(s_new)
ss_add.append(s_new)
ss += ss_add
print([(s, sum(s)) for s in ss if len(s) >= max_len])
import mylib, math
pp = sorted(mylib.primes_from_file(10**6 + 1))
ends = dict((x, [x * y % 10 for y in range(10)]) for x in [1, 3, 7, 9])
# print(ends)
ends_map = dict()
for p in ends:
ends_map[p] = [0 for x in range(10)]
for i in range(len(ends[p])):
ends_map[p][ends[p][i]] = i
print(ends_map)
def divide(x, y):
x_str = str(x)
z = x * 1
pwr = len(x_str)
d = 0
e = y % 10
print('*' + str(x), '/', y, end=' ')
for c in range(pwr):
n = z % 10
m = ends_map[e][n]
d += m * 10**c
z = (z - m * y) // 10
print(' > ', n, m, m * y, end=' ')
print(' : ', d * y, '/', y, '=', d)
return d * y