diff --git a/maths/allocation_number.py b/maths/allocation_number.py
index 4e74bb2e6..d419e74d0 100644
--- a/maths/allocation_number.py
+++ b/maths/allocation_number.py
@@ -1,37 +1,28 @@
+"""
+In a multi-threaded download, this algorithm could be used to provide
+each worker thread with a block of non-overlapping bytes to download.
+For example:
+    for i in allocation_list:
+        requests.get(url,headers={'Range':f'bytes={i}'})
+"""
 from __future__ import annotations
 
 
 def allocation_num(number_of_bytes: int, partitions: int) -> list[str]:
     """
     Divide a number of bytes into x partitions.
+    :param number_of_bytes: the total of bytes.
+    :param partitions: the number of partition need to be allocated.
+    :return: list of bytes to be assigned to each worker thread
 
-    In a multi-threaded download, this algorithm could be used to provide
-    each worker thread with a block of non-overlapping bytes to download.
-    For example:
-        for i in allocation_list:
-            requests.get(url,headers={'Range':f'bytes={i}'})
-
-    parameter
-    ------------
-    : param number_of_bytes
-    : param partitions
-
-    return
-    ------------
-    : return: list of bytes to be assigned to each worker thread
-
-    Examples:
-    ------------
     >>> allocation_num(16647, 4)
-    ['0-4161', '4162-8322', '8323-12483', '12484-16647']
-    >>> allocation_num(888, 888)
-    Traceback (most recent call last):
-        ...
-    ValueError: partitions can not >= number_of_bytes!
+    ['1-4161', '4162-8322', '8323-12483', '12484-16647']
+    >>> allocation_num(50000, 5)
+    ['1-10000', '10001-20000', '20001-30000', '30001-40000', '40001-50000']
     >>> allocation_num(888, 999)
     Traceback (most recent call last):
         ...
-    ValueError: partitions can not >= number_of_bytes!
+    ValueError: partitions can not > number_of_bytes!
     >>> allocation_num(888, -4)
     Traceback (most recent call last):
         ...
@@ -39,16 +30,16 @@ def allocation_num(number_of_bytes: int, partitions: int) -> list[str]:
     """
     if partitions <= 0:
         raise ValueError("partitions must be a positive number!")
-    if partitions >= number_of_bytes:
-        raise ValueError("partitions can not >= number_of_bytes!")
+    if partitions > number_of_bytes:
+        raise ValueError("partitions can not > number_of_bytes!")
     bytes_per_partition = number_of_bytes // partitions
-    allocation_list = [f"0-{bytes_per_partition}"]
-    for i in range(1, partitions - 1):
-        length = f"{bytes_per_partition * i + 1}-{bytes_per_partition * (i + 1)}"
-        allocation_list.append(length)
-    allocation_list.append(
-        f"{(bytes_per_partition * (partitions - 1)) + 1}-" f"{number_of_bytes}"
-    )
+    allocation_list = []
+    for i in range(partitions):
+        start_bytes = i * bytes_per_partition + 1
+        end_bytes = (
+            number_of_bytes if i == partitions - 1 else (i + 1) * bytes_per_partition
+        )
+        allocation_list.append(f"{start_bytes}-{end_bytes}")
     return allocation_list