A027895 -id:A027895 - cp's OEIS frontend

A226477 Table (read by rows) of the natural numbers (in ascending order) whose reciprocals have only periodic decimals of length k.

1, 3, 9, 11, 33, 99, 27, 37, 111, 333, 999, 101, 303, 909, 1111, 3333, 9999, 41, 123, 271, 369, 813, 2439, 11111, 33333, 99999, 7, 13, 21, 39, 63, 77, 91, 117, 143, 189, 231, 259, 273, 297, 351, 407, 429, 481, 693, 777, 819, 1001, 1221, 1287, 1443, 2079, 2331, 2457, 2849, 3003, 3367, 3663, 3861, 4329, 5291, 6993, 8547, 9009, 10101, 10989, 12987, 15873, 25641, 27027, 30303, 37037, 47619, 76923, 90909, 111111, 142857, 333333, 999999

Offset: 1

Martin Renner, Jun 08 2013

The k-th row always ends with 10^k - 1 = 99..99 (k times 9).

The number of elements in row k is A059892(k).

			The table T(k,m), m = 1..A059892(k), begins
  1, 3, 9;
  11, 33, 99;
  27, 37, 111, 333, 999;
  etc.

Jianing Song, Rows n = 1..32, flattened

Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895, A027896, A027897, A109933, A106305, A111117, A111211, A113116, A113522 (Divisors of 10^k - 1, k = 2..18), A059892, A084680.

a:=[1,3,9]: S:={1,3,9}: for k from 2 to 6 do T:=numtheory[divisors](10^k-1): a:=[op(a),op(T minus S)]: S:=S union T; od: a;

Row(n) = my(v=divisors(10^n-1)); select(x->(znorder(Mod(10,x))==n), v) \\ Jianing Song, Jun 15 2021

A197318 Divisors of the repunit 111111111111 = A002275(12).

Original entry on oeis.org

1, 3, 7, 11, 13, 21, 33, 37, 39, 77, 91, 101, 111, 143, 231, 259, 273, 303, 407, 429, 481, 707, 777, 1001, 1111, 1221, 1313, 1443, 2121, 2849, 3003, 3333, 3367, 3737, 3939, 5291, 7777, 8547, 9191, 9901, 10101, 11211, 14443, 15873, 23331, 26159, 27573, 29703

Offset: 1

M. F. Hasler, Oct 13 2011

The sequence is marked "full" since even though they don't fit into the three lines above, all 128 terms are known and available in the b-file or using the given PARI code.

M. F. Hasler, Table of n, a(n) for n = 1..128 (complete sequence)
Index entries for sequences related to divisors of numbers.

Cf. A109492, A197308, A197309, A070529, A018282, A018766, A027890-A027895, A027889.

Divisors[111111111111] (* Paolo Xausa, Jul 04 2024 *)

PARI
```
divisors(1e12\9)
```

A345319 Numbers whose reciprocals have period 10.

Original entry on oeis.org

451, 1353, 2981, 4059, 8943, 9091, 26829, 27273, 81819, 100001, 122221, 300003, 366663, 372731, 900009, 1099989, 1118193, 2463661, 3354579, 4100041, 7390983, 12300123, 22172949, 27100271, 36900369, 81300813, 101010101, 243902439, 303030303, 909090909, 1111111111, 3333333333, 9999999999

Offset: 1

Tanya Khovanova, Jun 13 2021

Equivalently, these are numbers k such that the multiplicative order of 10 modulo k is 10.
These are indices of terms at which 10 appears in A084680.
There are exactly A059892(10) = mu(10/10)*d(10^10-1) + mu(10/5)*d(10^5-1) + mu(10/2)*d(10^2-1) + mu(10/1)*d(10^1-1) = 48 - 12 - 6 + 3 = 33 terms, where d = A000005 and mu = A008683. - Jianing Song, Jun 15 2021

			1/451 = 0.00221729490022172949002217294900..., whose periodic part is 0022172949.

Cf. A084680, A059892.
Subsequence of A027895.
10th row of A226477.

Select[Range[100000000], MultiplicativeOrder[10, #] == 10 &]

isok(k) = gcd(k, 10) && (znorder(Mod(10, k)) == 10); \\ Michel Marcus, Jun 14 2021

my(v=divisors(10^10-1)); select(x->(znorder(Mod(10,x))==10), v) \\ Jianing Song, Jun 15 2021

a(27)-a(28) from Jinyuan Wang, Jun 13 2021

a(29)-a(33) from Jianing Song, Jun 15 2021

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A226477 Table (read by rows) of the natural numbers (in ascending order) whose reciprocals have only periodic decimals of length k.

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A197318 Divisors of the repunit 111111111111 = A002275(12).

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A345319 Numbers whose reciprocals have period 10.

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