cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A248956 Number of polynomials a_k*x^k + ... + a_1*x + a_0 with k > 0, integer coefficients and only non-multiple positive integer roots and a_0 = p^n (p is a prime).

Original entry on oeis.org

1, 3, 5, 9, 13, 19, 27, 37, 49, 65, 85, 109, 139, 175, 219, 273, 337, 413, 505, 613, 741, 893, 1071, 1279, 1523, 1807, 2137, 2521, 2965, 3477, 4069, 4749, 5529, 6425, 7449, 8619, 9955, 11475, 13203, 15167, 17393, 19913, 22765, 25985, 29617, 33713, 38321, 43501
Offset: 0

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Author

Reiner Moewald, Oct 17 2014

Keywords

Comments

If D_n = {p^0, ..., p^n} is the set of all positive divisors of p^n (p is a prime), then a(n) gives the number of all subsets of D_n for which the product of all their elements is a divisor of p^n. Furthermore, a(n) gives the number of all strict partitions of n including the integer 0.

Examples

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

Crossrefs

Partial sums of A087135.

Formula

a(n) = -1 + 2*Sum_{k=0..n} a*(k) where a*(n) = A000009(n).
a(n) = A248955(p^n), where p is any prime. - Michel Marcus, Nov 07 2014
a(n) = 2*A036469(n) - 1. - Hiroaki Yamanouchi, Nov 21 2014

Extensions

a(20)-a(22) from Michel Marcus, Nov 07 2014
a(23)-a(47) from Hiroaki Yamanouchi, Nov 21 2014

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

Original entry on oeis.org

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

Views

Author

Reiner Moewald, Oct 06 2014

Keywords

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.
		

Crossrefs

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")
Showing 1-2 of 2 results.