Add cube counting functions and examples

cube_count.py

@@ -0,0 +1,33 @@
+def int_cuberoot_floor(x):
+    """Return floor(cuberoot(x)) for nonnegative integer x using binary search."""
+    if x < 0:
+        raise ValueError("x must be nonnegative")
+    lo, hi = 0, int(x**(1/3)) + 2  # initial bracket (safe)
+    # make hi definitely large enough
+    while hi**3 <= x:
+        hi *= 2
+    while lo + 1 < hi:
+        mid = (lo + hi) // 2
+        if mid**3 <= x:
+            lo = mid
+        else:
+            hi = mid
+    return lo
+
+def count_pairs(n):
+    """Return count of pairs (a>=1, b>=0) of integers with a^3 + b^3 = n,
+       and also return the list of such pairs."""
+    pairs = []
+    max_a = int_cuberoot_floor(n)  # a^3 <= n
+    for a in range(1, max_a + 1):
+        rem = n - a**3
+        if rem < 0:
+            break
+        b = int_cuberoot_floor(rem)
+        if b**3 == rem:
+            pairs.append((a, b))
+    return len(pairs), pairs
+
+# Examples
+print(count_pairs(9))   # -> (2, [(1,2),(2,1)])
+print(count_pairs(27))  # -> (1, [(3,0)])