A248410 a(n) = number of polynomials a_k*x^k + ... + a_1*x + n with k > 0, integer coefficients and only distinct integer roots.

3, 11, 11, 23, 11, 43, 11, 47, 23, 43, 11, 103, 11, 43, 43, 83, 11, 103, 11, 103, 43, 43, 11, 223, 23, 43, 47, 103, 11, 187, 11, 139, 43, 43, 43, 275, 11, 43, 43, 223, 11, 187, 11, 103, 103, 43, 11, 427, 23, 103, 43, 103, 11, 223, 43, 223, 43, 43, 11, 503, 11, 43, 103, 227, 43, 187, 11, 103, 43, 187, 11, 635, 11, 43, 103, 103, 43, 187, 11

Offset: 1

Author

Reiner Moewald, Oct 06 2014

Keywords

• nonn

Comments

If D_n is the set of all positive and negative divisors of n, then a(n) is the number of all subsets of D_n for which the product of all their elements is a divisor of n. a(n) depends only on the prime signature of n.

Examples

			a(1)=3: x + 1; -x + 1; -x^2 + 1.
		

Links

• Reiner Moewald, Table of n, a(n) for n = 1..502

Crossrefs

Cf. A248348, A248955.

Programs

Python

from itertools import chain, combinations
def powerset(iterable):
   s = list(iterable)
   return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))
print("Start")
a_n = 0
for num in range(1,1000):
   div_set = set((-1,1))
   a_n = 0
   for divisor in range(1, num + 1):
      if (num % divisor == 0):
         div_set.add(divisor)
         div_set.add(divisor*(-1))
   pow_set = set(powerset(div_set))
   num_set = len(pow_set)
   for count_set in range(0, num_set):
      subset = set(pow_set.pop())
      num_subset = len(subset)
      prod = 1
      if num_subset < 1:
         prod = 0
      for count_subset in range (0, num_subset):
         prod = prod * subset.pop()
      if prod != 0:
         if (num % prod == 0):
            a_n = a_n +1
   print(num, a_n)
print("Ende")