matplotlib 1.5+

You can use axes.set_prop_cycle (example).

matplotlib 1.0-1.4

You can use axes.set_color_cycle (example).

matplotlib 0.x

You can use Axes.set_default_color_cycle.

Because your num_values spans an exponential range, the colors for the values in the middle aren't being mapped correctly. It's not clear to me what your end goal is for this colormap, but if all you want to do is plot the colors and have the x-ticks show the corresponding number, you can do the following:

fig, ax = plt.subplots(figsize=(10,2))
ax.imshow((range(len(num_values)),),  cmap=ListedColormap(hex_values))
ax.set_xticks(range(len(num_values)), num_values, rotation=90)
fig.tight_layout()
fig.show()

Often, there is no need to get the default color cycle from anywhere, as it is the default one, so just using it is sufficient.

import numpy as np
import matplotlib.pyplot as plt

fig = plt.figure()
ax = fig.add_subplot(111)

t = np.arange(5)

for i in range(4):
    line, = ax.plot(t,i*(t+1), linestyle = '-')
    ax.plot(t,i*(t+1)+.3,color = line.get_color(), linestyle = ':')

plt.show()

In case you want to use the default color cycle for something different, there are of course several options.

"tab10" colormap

First it should be mentionned that the "tab10" colormap comprises the colors from the default color cycle, you can get it via cmap = plt.get_cmap("tab10").

Equivalent to the above would hence be

import numpy as np
import matplotlib.pyplot as plt

fig = plt.figure()
ax = fig.add_subplot(111)

t = np.arange(5)
cmap = plt.get_cmap("tab10")
for i in range(4):
    ax.plot(t,i*(t+1),   color=cmap(i), linestyle = '-')
    ax.plot(t,i*(t+1)+.3,color=cmap(i), linestyle = ':')

plt.show()

Colors from color cycle

You can also use the color cycler directly, cycle = plt.rcParams['axes.prop_cycle'].by_key()['color']. This gives list with the colors from the cycle, which you can use to iterate over.

import numpy as np
import matplotlib.pyplot as plt

fig = plt.figure()
ax = fig.add_subplot(111)

t = np.arange(5)
cycle = plt.rcParams['axes.prop_cycle'].by_key()['color']

for i in range(4):
    ax.plot(t,i*(t+1),   color=cycle[i], linestyle = '-')
    ax.plot(t,i*(t+1)+.3,color=cycle[i], linestyle = ':')

plt.show()

The CN notation

Finally, the CN notation allows to get the Nth color of the color cycle, color="C{}".format(i). This however only works for the first 10 colors (N in [0,1,...9])

import numpy as np
import matplotlib.pyplot as plt

fig = plt.figure()
ax = fig.add_subplot(111)

t = np.arange(5)

for i in range(4):
    ax.plot(t,i*(t+1),   color="C{}".format(i), linestyle = '-')
    ax.plot(t,i*(t+1)+.3,color="C{}".format(i), linestyle = ':')

plt.show()

All codes presented here produce the same plot.