Question

I am working on Project Euler #35, and I need to find the circular permutations of a number. Using itertools, I can easily get the permutations of a number. However, I want to do it with a list comprehension (as it seems more Pythonic; I am also trying to get familiar with list comprehensions).

I found that all circular primes can only contain the digits 1, 3, 7, and 9 (this excludes 2 and 5, which are circular primes by definition). If any other digit was in the number (0, 2, 4, 5, 6, or 8) one of the permutations would not be a prime (as that digit would be last in at least one of the permutations).

Thus, I tried doing this:

from itertools import permutations
l = [x for x in list(permutations('1397', y)) for y in range(7)]

I needed to use y for y in range(7) so that I get varying lengths of permutations.

However, this gave me a TypeError:

Traceback (most recent call last):
  File "<pyshell#23>", line 1, in <module>
    l = [x for x in list(permutations('1397', y)) for y in range(7)]
TypeError: an integer is required

This works, but it isn’t using two variables in one list comprehension:

l = []
for y in range(7):
    l.append([x for x in list(permutations('1379', y))])

How can I do a double-variable list comprehension? Thanks!

Answer 1

The for y in range(7) part should come before the permutation loop.:

l = [x for y in range(7) for x in list(permutations('1397', y))]

The above list comprehension is equivalent to :

In [93]: l = []

In [94]: for y in range(7):
    ...:     l.extend(list(permutations('1397', y)))

For example:

In [76]: l = [x for y in range(3) for x in list(permutations('1397', y))]

In [77]: l
Out[77]: 
[(),
 ('1',),
 ('3',),
 ('9',),
 ('7',),
 ('1', '3'),
 ('1', '9'),
 ('1', '7'),
 ('3', '1'),
 ('3', '9'),
 ('3', '7'),
 ('9', '1'),
 ('9', '3'),
 ('9', '7'),
 ('7', '1'),
 ('7', '3'),
 ('7', '9')]

And the list-comprehension version for your working example,

l = []
for y in range(7):
    l.append(list(permutations('1397', y)))

is:

In [85]: l = [list(permutations('1397', y)) for y in range(3)]

In [86]: l
Out[86]: 
[[()],
 [('1',), ('3',), ('9',), ('7',)],
 [('1', '3'),
  ('1', '9'),
  ('1', '7'),
  ('3', '1'),
  ('3', '9'),
  ('3', '7'),
  ('9', '1'),
  ('9', '3'),
  ('9', '7'),
  ('7', '1'),
  ('7', '3'),
  ('7', '9')]]