##########################################################################
#  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 A111776(n):
    r"""
    This function returns the $n$-th number of sequence A111776

    $n$ is the sum of at most $a(n)$ consecutive positive integers.
    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 $d$ of $n$ there is a unique corresponding
    $k = min(d,2n/d)$. $a(n)$ is the largest among those $k$
.
    See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

    INPUT:
        n -- non negative integer

    OUTPUT:
        integer -- function value

    EXAMPLES:
        sage: A111776(15)
        5

        sage: A111776(1000)
        25

        sage: A111776(51)
        6

        sage: A111776(23)
        2

        sage: A111776(1)
        1

        AUTHOR:
            - Jaap Spies (2006-12-08)
    """
    if n == 1 or n == 0:
        return 1
    m = 0
    for d in [i for i in divisors(n) if i%2]: # d is odd divisor
        k = min(d, 2*n/d)
        if k > m:
            m = k
    return m

def A111776_list(N):
     return [A111776(i) for i in range(0,N+1)]