A052038 First nonzero digit in expansion of 1/n.

1, 5, 3, 2, 2, 1, 1, 1, 1, 1, 9, 8, 7, 7, 6, 6, 5, 5, 5, 5, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 9, 9, 9, 9

Offset: 1

Author

Patrick De Geest, Dec 15 1999

Keywords

• nonn
• base
• easy

Comments

The number of times each digit occurs for numbers < 10^k:

...\a(n)==1.........2.......3........4........5........6........7........8........9

10^k\

1.........5.........2........1........0........1........0........0........0........0

2........55........19........9........5........5........2........2........1........1

3.......555.......186.......92.......55.......39.......26.......19.......15.......12

4......5555......1853......925......555......373......264......197......154......123

5.....55555.....18520.....9258.....5555.....3707.....2645.....1982.....1543.....1234

6....555556....185187....92591....55555....37041....26454....19839....15432....12345

7...5555555...1851854...925924...555555...370375...264549...198410...154321...123456

8..55555555..18518521..9259257..5555555..3703709..2645501..1984124..1543210..1234567

9.555555555.185185188.92592590.55555555.37037043.26455025.19841266.15432099.12345678

...

Inf. ...5/9......5/27.....5/54.....5/90.....1/27........?........?........?........?

Crossrefs

Cf. A052039, A033330, A033420, A061861.

Programs

Mathematica

f[n_] := RealDigits[1/n, 10, 12][[1, 1]]; Array[f, 105]

Formula

a(n) = floor(10^floor(1+log_10(n-1))/n). After 10^k terms the number of times m will have appeared will be about 10^(k+1)/(9*m*(m+1)), e.g., 1 will appear just over 55.5% of the time. - Henry Bottomley, May 11 2001

a(n) = A000030(floor(A011557(k)/n)) for k >= A004218(n). - Reinhard Zumkeller, Feb 27 2011