A059892 -id:A059892 - cp's OEIS frontend

A059910 a(n) = |{m : multiplicative order of n mod m = 5}|.

0, 1, 4, 6, 9, 4, 4, 6, 20, 9, 8, 2, 6, 6, 12, 44, 5, 6, 18, 14, 12, 4, 4, 2, 56, 13, 20, 4, 6, 2, 40, 6, 18, 12, 12, 44, 63, 6, 28, 4, 16, 14, 8, 2, 18, 12, 28, 14, 70, 3, 42, 12, 42, 6, 24, 8, 56, 44, 60, 6, 60, 2, 4, 90, 21, 20, 24, 2, 18, 60, 88, 6, 12, 2, 28, 26, 6, 28, 8, 14, 170

Offset: 1

Vladeta Jovovic, Feb 08 2001

The multiplicative order of a mod m, gcd(a,m) = 1, is the smallest natural number d for which a^d = 1 (mod m).

Cf. A059907-A059909, A059911, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

a[n_] := Subtract @@ DivisorSigma[0, {n^5-1, n-1}]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 25 2025 *)

a(n) = if(n == 1, 0, numdiv(n^5-1) - numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^5-1)-tau(n-1), where tau(n) = number of divisors of n A000005. Generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A059910 a(n) = |{m : multiplicative order of n mod m = 5}|.

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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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A345319 Numbers whose reciprocals have period 10.

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