def first_triangular_with_k_divisors(k, lo=2, hi=10**3, pt=None):
'''
---------------------------------------------------------------------------
first_triangular_with_k_divisors
---------------------------------------------------------------------------
Returns the first triangular number that has k divisors.
Searches in the range [lo, hi) initially, if it fails to find a result,
then it doubles the search range by 2 (i.e. lo <- hi, hi <- 2*hi) and
searches there. You can optionally provide lo and hi to start the search,
otherwise, these default to 2 and 10**3.
Uses a PrimeTools object to compute divisors, which can optionally be
provided. If not provided, will use PrimeTools(hi * (hi+1) / 2).
'''
if pt is None:
pt = PrimeTools(triangle(hi))
else:
pt.extend(triangle(hi))
for i in range(lo, hi):
tri = triangle(i)
n_divisors = pt.n_divisors(tri)
if n_divisors > k:
return tri
else:
return first_triangular_with_k_divisors(k, lo=hi, hi=2 * hi, pt=pt)
def find_nth_prime(n, pt=None):
if pt is None:
# if no primetools object has been provided, then
# begin by finding all primes below 100
m = 100
pt = PrimeTools(m)
# keep extending prime list until we have n primes
while len(pt.prime_list) < n:
m = 2 * m
pt.extend(m)
return pt.prime_list[n - 1]
def sum_primes_below(n, pt=None):
'''
---------------------------------------------------------------------------
sum_primes_below
---------------------------------------------------------------------------
Sum primes below n, excluding n itself. I.e. it returns:
sum([p : p prime, p < n])
You can optionally provide a pre-computed PrimeTools object (pt) whose
sieve will be extended if necessary.
'''
if pt is None:
pt = PrimeTools(n)
else:
pt.extend(n)
return sum([p for p in pt.prime_list if p < n])
