
A   taxicab number   (the definition that is being used here)   is a positive integer that can be expressed as the sum of two positive cubes in more than one way.



The first taxicab number is   1729,   which is:

 13   +   123       and also
 93   +   103.


Taxicab numbers are also known as:

   taxi numbers
   taxi-cab numbers
   taxi cab numbers
   Hardy-Ramanujan numbers


Task

 Compute and display the lowest 25 taxicab numbers (in numeric order, and in a human-readable format).
 For each of the taxicab numbers, show the number as well as it's constituent cubes.


Extra credit
 Show the 2,000th taxicab number, and a half dozen more


See also

 A001235: taxicab numbers on The On-Line Encyclopedia of Integer Sequences.
 Hardy-Ramanujan Number on MathWorld.
 taxicab number on MathWorld.
 taxicab number on Wikipedia   (includes the story on how taxi-cab numbers came to be called).


from collections import defaultdict
from itertools import product
from pprint import pprint as pp
 
cube2n = {x**3:x for x in range(1, 1201)}
sum2cubes = defaultdict(set)
for c1, c2 in product(cube2n, cube2n):
	if c1 >= c2: sum2cubes[c1 + c2].add((cube2n[c1], cube2n[c2]))
 
taxied = sorted((k, v) for k,v in sum2cubes.items() if len(v) >= 2)
 
#pp(len(taxied))  # 2068
for t in enumerate(taxied[:25], 1):
    pp(t)
print('...')    
for t in enumerate(taxied[2000-1:2000+6], 2000):
    pp(t)