##########################################################################
#  Copyright (C) 2006 Jaap Spies, jaapspies@gmail.com
#  Copyright (C) 2006 William Stein, wstein@gmail.com
#
#  Distributed under the terms of the GNU General Public License (GPL):
#
#                  http://www.gnu.org/licenses/
###############################################################

def A111775(n):
    r"""
    This function returns the $n$-th number of Sloane's sequence A111775

    Powers of 2 and (odd) primes can not be written as a sum of at least
    three consecutive integers. $a(n)$ strongly depends on the number
    of odd divisors of $n$ (A001227):
    Suppose $n$ is to be written as sum of $k$ consecutive integers
    starting with $m$, then $2n = k(2m + k - 1)$.
    Only one of the factors is odd. For each odd divisor of $n$
    there is a unique corresponding $k$, $k=1$ and $k=2$ must be excluded.

    See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

    INPUT:
        n -- non negative integer

    OUTPUT:
        integer -- function value

    EXAMPLES:
        sage: A111775(15)
        2

        sage: A111775(1000)
        3

        sage: A111775(51)
        2

        sage: A111775(23)
        0

        sage: A111775(1)
        0

        sage: A111775(0)
        0

        AUTHOR:
            - Jaap Spies (2006-12-09)
    """
    if n == 1 or n == 0:
        return 0
    k = sum(i%2 for i in divisors(n)) # A001227, the number of odd divisors
    if n % 2 ==0:
        return k-1
    else:
        return k-2

def A111775_list(N):
     return [A111775(i) for i in range(0,N+1)]