A052039 -id:A052039 - cp's OEIS frontend

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

Patrick De Geest, Dec 15 1999

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........?........?........?........?

Cf. A052039, A033330, A033420, A061861.

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

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

A259228 Lexicographically earliest sequence of distinct terms such that the first significant digits of 1/n start with a(n).

Original entry on oeis.org

1, 5, 3, 2, 20, 16, 14, 12, 11, 10, 9, 8, 7, 71, 6, 62, 58, 55, 52, 50, 4, 45, 43, 41, 40, 38, 37, 35, 34, 33, 32, 31, 30, 29, 28, 27, 270, 26, 25, 250, 24, 23, 232, 22, 222, 21, 212, 208, 204, 200, 19, 192, 18, 185, 181, 17, 175, 172, 169, 166, 163, 161, 15

Offset: 1

Paul Tek, Jun 21 2015

This is a permutation of the positive integers.

All fixed points are in A065924.

			+----+---------+----+
|  n |   1/n   |a(n)|
+----+---------+----+
|  1 | 1       |  1 |
|  2 | 0.5...  |  5 |
|  3 | 0.3...  |  3 |
|  4 | 0.2...  |  2 |
|  5 | 0.20... | 20 |
|  6 | 0.16... | 16 |
|  7 | 0.14... | 14 |
|  8 | 0.12... | 12 |
|  9 | 0.11... | 11 |
| 10 | 0.10... | 10 |
+----+---------+----+

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, PARI program for this sequence
Index entries for sequences that are permutations of the natural numbers

Cf. A052039, A065924, A259229.

PARI
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See Link section.
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A099406 Decimal part of 1/a(n) starts with the n-th prime (leading zeros excluded).

Original entry on oeis.org

4, 3, 2, 13, 9, 72, 56, 51, 42, 34, 32, 27, 24, 23, 21, 186, 167, 162, 148, 14, 136, 126, 12, 112, 103, 99, 97, 93, 91, 88, 782, 76, 725, 715, 67, 66, 633, 61, 596, 575, 556, 55, 521, 516, 506, 501, 472, 447, 44, 435, 428, 417, 414, 397, 388, 38, 371, 368, 36, 355, 353

Offset: 1

Gil Broussard, Nov 17 2004

			a(1)= 4 -> 1/4 =0.{2}500000...
a(2)= 3 -> 1/3 =0.{3}333333...
a(3)= 2 -> 1/2 =0.{5}000000...
a(4)=13 -> 1/13=0.0{7}69230...
a(100)=1846 -> 1/1846=0.000{541}712 and 541 is the 100th prime.

Cf. A034057, A052039.

a(n) = A052039(prime(n)). - Michel Marcus, Jan 08 2025

A326818 a(n) is the smallest k such that the first significant digits of 1/k coincide with n.

Original entry on oeis.org

1, 4, 3, 21, 2, 15, 13, 12, 11, 1, 9, 8, 72, 7, 63, 6, 56, 53, 51, 5, 46, 44, 42, 41, 4, 38, 36, 35, 34, 33, 32, 31, 3, 29, 28, 271, 27, 26, 251, 25, 24, 233, 23, 223, 22, 213, 21, 205, 201, 2, 193, 19, 186, 182, 18, 176, 173, 17, 167, 164, 162, 16, 157, 154, 152

Offset: 1

Giovanni Resta, Oct 20 2019

This sequence differs from A052039 in how it treats reciprocals with terminating representation, i.e., the values 1/k for integers k whose prime factors are 2 and/or 5. For example, here we assume 1/5 = 0.2000... which leads to a(20) = 5, while in A052039 we consider 1/5 = 0.2 (without trailing zeros), which leads to A052039(20) = 48 instead.

			a(123) = 81 because 1/81 = 0.0(123)4... and 81 is the smallest number with this property.

Cf. A052039, A034057.

a[n_] := Block[{d = IntegerDigits[n], m, k = 1}, m = Length[d]; While[ RealDigits[1/k, 10, m][[1]] != d, k++]; k]; Array[a, 65]

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A052038 First nonzero digit in expansion of 1/n.

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A259228 Lexicographically earliest sequence of distinct terms such that the first significant digits of 1/n start with a(n).

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PARI

A099406 Decimal part of 1/a(n) starts with the n-th prime (leading zeros excluded).

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A326818 a(n) is the smallest k such that the first significant digits of 1/k coincide with n.

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