##########################################################################
#  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 A111774(n):
    r"""
    This function returns the $n$-th number of Sloane's sequence A111774

    """
    a = A111774_list(150)
    return a[n-1]

def A111774_list(n):
    return [i for i in range(1, n+1) if is_number_of_the_third_kind(i)]

def is_number_of_the_third_kind(n):
    r""""
    This function returns True iff $n$ is a number of the third kind.

    A number of the third kind can be written as a sum of at least
    three consecutive positive integers.
    Odd primes can only be written as a sum of two consecutive integers.
    Powers of 2 do not have a representation as a sum of $k$ consecutive
    integers (other than the trivial $n = n$ for $k = 1$).

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

    INPUT:
        n -- positive integer

    OUTPUT:
        True -- if n is not prime and not a power of 2
        False --

    EXAMPLES:
        sage: is_number_of_the_third_kind(6)
        True

        sage: is_number_of_the_third_kind(100)
        True

        sage: is_number_of_the_third_kind(16)
        False

        sage: is_number_of_the_third_kind(97)
        False

    AUTHOR:
        - Jaap Spies (2006-12-09)
    """

    if (not is_prime(n)) and (not is_power_of_two(n)):
        return True
    else:
        return False


def is_power_of_two(n):
    r""""
    This function returns True iff $n$ is a power of 2

    INPUT:
        n -- integer

    OUTPUT:
        True -- if n is a power of 2
        False -- if not

    EXAMPLES:
        sage: is_power_of_two(1024)
        True

        sage: is_power_of_two(1)
        True

        sage: is_power_of_two(24)
        False

        sage: is_power_of_two(0)
        False

        sage: is_power_of_two(-4)
        False

    AUTHOR:
        - Jaap Spies (2006-12-09)

    """
# modification of is2pow(n) from the Programming Guide
    while n > 0 and n%2 == 0: 
        n = n >> 1
    return n == 1