一百万的数位和

把从 0 到 999,999 的所有数都看成带前导零的 6 位数。这样一共有 $$10^6 = 1{,}000{,}000$$ 个数。

对任意一个数位位置来说，数字 $0,1,\dots,9$ 都恰好各出现 100,000 次，因此该位置对总数位和的贡献是 $$(0+1+\cdots+9)\cdot100{,}000 = 45\cdot100{,}000 = 4{,}500{,}000.$$

一共有 6 个数位，所以从 0 到 999,999 的总数位和为 $$6\cdot4{,}500{,}000 = 27{,}000{,}000.$$

题目要的是从 1 到 1,000,000（含），因此只需再加上 $1,000,000$ 的数位和 $1$，而数字 0 的数位和是 0，不影响结果。

所以答案是 $$27{,}000{,}000 + 1 = \boxed{27{,}000{,}001}.$$

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Original Explanation

It’s often easier to count digit contributions.

Every integer from 0 to 999,999 is a 6-digit string if we allow leading zeros. So in that range we have exactly $10^6 =1,000,000$ numbers, each with 6 digits.

Each of the 6 positions (hundreds of thousands, ten-thousands, …, ones) cycles evenly through 0-9.

• In each position, each digit 0,1,…,9 appears exactly 100,000 times (since 1,000,000 / 10 = 100,000).

• So the total contribution of that digit position to the digit sum is: $$(0+1+⋯+9)⋅100,000=45⋅100,000=4,500,000.$$

• With 6 digit positions, the grand total from 0 to 999,999 is $$6⋅4,500,000=27,000,000.$$

We want 1 to 1,000,000 not 0 to 999,999

• If we take the sum from 0 to 999,999 we get 27,000,000
• Add the digit sum of 1,000,000 which is 1.
• Remove the digit sum of 0, which is 0.

So the final answer is:

$$27,000,000 + 1 = \boxed{27,000,001}$$