Problem 284
Steady Squares
The 3-digit number 376 in the decimal numbering system is an example of numbers with the special property that its square ends with the same digits: 3762 = 141376. Let’s call a number with this property a steady square.
Steady squares can also be observed in other numbering systems. In the base 14 numbering system, the 3-digit number c37 is also a steady square: c372 = aa0c37, and the sum of its digits is c+3+7=18 in the same numbering system. The letters a, b, c and d are used for the 10, 11, 12 and 13 digits respectively, in a manner similar to the hexadecimal numbering system.
For 1 ≤ n ≤ 9, the sum of the digits of all the n-digit steady squares in the base 14 numbering system is 2d8 (582 decimal). Steady squares with leading 0’s are not allowed.
Find the sum of the digits of all the n-digit steady squares in the base 14 numbering system for 1 ≤ n ≤ 10000 (decimal) and give your answer in the base 14 system using lower case letters where necessary.
自守数
十进制表示的三位数376拥有一种特殊的性质，它的平方的末尾就是它本身：3762 = 141376。我们称拥有这种性质的数为自守数。
在其它进制下也存在自守数。在14进制下，三位数c37同样是自守数：c372 = aa0c37，它的各位数字和在14进制下是c+3+7=18。注意这里使用的字母a、b、c、d分别代表10、11、12、13，和16进制下相似。
对于1 ≤ n ≤ 9，所有14进制下的n位自守数的各位数字和为2d8（转化成十进制数为582）。自守数不允许有前导零。
对于1 ≤ n ≤ 10000（十进制数），求所有14进制下的n位自守数的各位数字和，将你的答案用14进制表示，并在有需要时使用小写字母。