# 数字和

## 定義

${\displaystyle F_{b}(n)=\sum _{i=0}^{k-1}d_{i}}$

${\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b}}^{i}}{b^{i}}}}$

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10...（OEIS數列A007953

## 列表

1 1, 10, 100, 1000, 10000, 100000, … A011557
2 2, 11, 20, 101, 110, 200, 1001, … A052216 A069537 2,11,101
3 3, 12, 21, 30, 102, 111, 120, 201, 210, 300, 1002, … A052217
4 4, 13, 22, 31, 40, 103, 112, 121, 130, 202, 211, … A052218 A063997 A062339
5 5, 14, 23, 32, 41, 50, 104, 113, 122, 131, 140, 203, A052219 A069540 A062341
6 6, 15, 24, 33, 42, 51, 60, 105, 114, 123, 132, 141, 150, … A052220 A062768
7 7, 16, 25, 34, 43, 52, 61, 70, 106, 115, 124, 133, 142, … A052221 A063416 A062337
8 8, 17, 26, 35, 44, 53, 62, 71, 80, 107, 116, 125, 134, … A052222 A069543 A062343
9 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 108, 117, 126, 135, … A052223
10 19, 28, 37, 46, 55, 64, 73, 82, 91, 109, 118, 127, 136, … A052224 A218292 A107579
11 29, 38, 47, 56, 65, 74, 83, 92, 119, 128, 137, 146, 155, … A166311 A216995 A106754
12 39, 48, 57, 66, 75, 84, 93, 129, 138, 147, 156, 165, … A235151

1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 199, 299, 399, …（OEIS數列A051885

## 相關數列

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, 2, 1.... （OEIS數列A274580

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 202, 211, 220.... （OEIS數列A193772

-2, -3, -4, -5, -6, -7, -8, -9, -15, -16, -17, -18, -19, -28, -29, -159, -168, -169, -178, -179, -187, -188, -189, -197, -198, -199。（OEIS數列A328933

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8...（OEIS數列A007954

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683.... （OEIS數列A256229

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, 2, 1...（OEIS數列A225693

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 2, 5, 10, 17, 26, 37, 50, 65...（OEIS數列A003132

## 參考文獻

1. ^ Bush, L. E., An asymptotic formula for the average sum of the digits of integers, American Mathematical Monthly (Mathematical Association of America), 1940, 47 (3): 154–156, JSTOR 2304217, doi:10.2307/2304217.
2. ^ Borwein, J. M.; Borwein, P. B., Strange series and high precision fraud (PDF), American Mathematical Monthly, 1992, 99 (7): 622–640 [2020-03-14], JSTOR 2324993, doi:10.2307/2324993, （原始内容存档 (PDF)于2016-05-09）.
3. Sloane, N.J.A. (编). Sequence A328933 (Zombie Numbers). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
4. Stu Savory. Divisibility by prime numbers under 50. 2003 [2020-03-14]. （原始内容存档于2019-09-16）.
5. ^ Weisstein, Eric W. (编). Casting Out Nines. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2020-03-14] （英语）.
6. ^ Bloch, R. M.; Campbell, R. V. D.; Ellis, M., The Logical Design of the Raytheon Computer, Mathematical Tables and Other Aids to Computation (American Mathematical Society), 1948, 3 (24): 286–295, JSTOR 2002859, doi:10.2307/2002859.
7. ^ Edgeworth, F. Y., The Mathematical Theory of Banking (PDF), Journal of the Royal Statistical Society, 1888, 51 (1): 113–127, （原始内容 (PDF)存档于2006-09-13）.
8. ^ David Applegate and Marc LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv, 2016-05-19 |year=|date=不匹配 (帮助)