def first_over(count, max_prime):
primes = sieve(max_prime)
i = 2
while count_sums(i, 0,
sorted(filter(lambda x: x <= i, primes))[::-1]) <= count:
i += 1
return i
def least_totient(max_):
primes = sieve(max_)
best_n = 0
best_tot = 1
for n in range(2,max_):
if n in primes:
continue
tot = totient(n,primes)
if n / tot > best_n / best_tot:
best_n = n
best_tot = tot
return best_n
def num_proper_fractions_2(max_):
primes = sieve(max_)
total = max_-1
for i in range(2,max_):
cur = 0
if i in primes:
total += (max_-i)-((max_-i)//i)
else:
if len(pf := prime_factors(i)) == 1:
total += (max_-i)-((max_-i)//next(iter(pf)))
else:
total += range_totient(i,min_=i,max_=max_,factors=pf)
def num_sums_below(sum_bound, prime_bound):
primes = sieve(prime_bound)
primes_2 = set(filter(lambda x: x <= sum_bound**(1 / 2), primes))
primes_3 = set(filter(lambda x: x <= sum_bound**(1 / 3), primes))
primes_4 = set(filter(lambda x: x <= sum_bound**(1 / 4), primes))
found = set()
for square in primes_2:
for cube in primes_3:
for quad in primes_4:
if (sum_ := square**2 + cube**3 + quad**4) < sum_bound:
found.add(sum_)
def least_totient_perm(max_):
primes = sieve(max_)
best_n = max_ + 1
best_tot = 1
for n in range(2, max_):
if n % 500000 == 0:
print(n)
if n in primes:
found[n] = n - 1
continue
tot = totient(n, primes)
if n / tot < best_n / best_tot:
if is_permutation(n, tot):
best_n = n
best_tot = tot
return best_n