A002371 -id:A002371 - cp's OEIS frontend

A385222 a(n) is the multiplicative order of -10 modulo prime(n); a(1) = a(3) = 0 for completion.

0, 2, 0, 3, 1, 3, 16, 9, 11, 28, 30, 6, 10, 42, 23, 26, 29, 60, 66, 70, 8, 26, 82, 44, 96, 4, 17, 106, 108, 112, 21, 65, 8, 23, 148, 150, 39, 162, 83, 86, 89, 180, 190, 192, 49, 198, 15, 111, 226, 228, 232, 14, 15, 25, 256, 131, 268, 10, 138, 28

Offset: 1

Jianing Song, Jun 27 2025

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A007348 (primes having primitive root -10).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, A380531, A380532, A380533, A380540, A380541, A380542, this sequence.

A385222[n_] := If[n == 1 || n == 3, 0, MultiplicativeOrder[-10, Prime[n]]];
Array[A385222, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-10}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

A007450 Decimal expansion of 1/17.

Original entry on oeis.org

0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5

Offset: 0

N. J. A. Sloane

Period 16: repeat [0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1, 1, 7, 6, 4, 7]. - Joerg Arndt, Mar 25 2013

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, 'Die periodischen Dezimalbrueche'. [From Reinhard Zumkeller, Oct 06 2008]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
R. K. Hoeflin, Mega Test
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,-1,1).

I:=[0, 5, 8, 8, 2, 3, 5, 2, 9]; [n le 9 select I[n] else Self(n-1)-Self(n-8)+Self(n-9): n in [1..100]]; // Vincenzo Librandi, Mar 25 2013

CoefficientList[Series[-x (7 x^7 - 3 x^6 + 2 x^5 + x^4 - 6 x^3 + 3 x + 5)/((x - 1) (x^8+1)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 25 2013 *)

a(n)=[0,5,8,8,2,3,5,2,9,4,1,1,7,6,4,7][n%16+1]; /* Joerg Arndt, Mar 25 2013 */

From Reinhard Zumkeller, Oct 06 2008: (Start)
A028416(4)=17; A002371(A049084(17)) = A002371(7)=16;
a(n+16) = a(n), a(n+16/2) = 9 - a(n). (End)
G.f.: -x*(7*x^7-3*x^6+2*x^5+x^4-6*x^3+3*x+5)/((x-1)*(x^8+1)). - Colin Barker, Aug 15 2012

A007615 Primes with unique period length (the periods are given in A007498).

Original entry on oeis.org

3, 11, 37, 101, 333667, 9091, 9901, 909091, 1111111111111111111, 11111111111111111111111, 99990001, 999999000001, 909090909090909091, 900900900900990990990991, 9999999900000001, 909090909090909090909090909091, 900900900900900900900900900900990990990990990990990990990991

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Additional terms are Phi(n,10)/gcd(n,Phi(n,10)) for the n in A007498, where Phi(n,10) is the n-th cyclotomic polynomial evaluated at 10.

			3 is the only prime p such that decimal expansion of 1/p has (nontrivial) period exactly 1.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Samuel Yates, Period Lengths of Exactly One or Two Prime Numbers, J. Rec. Math., 18 (1985), 22-24.

Max Alekseyev, Table of n, a(n) for n = 1..98 (terms 1..25 from T. D. Noe; terms 26..31 from Ray Chandler)
C. K. Caldwell, The Prime Glossary, unique prime
Makoto Kamada, Factorizations of Phi_n(10)
Index entries for sequences related to decimal expansion of 1/n

Cf. A007498, A040017, A002371, A048595, A006883, A007732, A051626, A061075, A006530, A019328.

nmax = 50; periods = Reap[ Do[ p = Cyclotomic[n, 10] / GCD[n, Cyclotomic[n, 10]]; If[ PrimeQ[p], Sow[n]], {n, 1, nmax}]][[2, 1]]; Cyclotomic[#, 10] / GCD[#, Cyclotomic[#, 10]]& /@ periods // Prepend[#, 3]& (* Jean-François Alcover, Mar 28 2013 *)

a(n) = A061075(A007498(n)). - Max Alekseyev, Oct 16 2010

a(n) = A006530(A019328(A007498(n))). - Ray Chandler, May 10 2017

A071126 Length of least repunit which is a multiple of the n-th prime, or 0 if no such multiple exists.

Original entry on oeis.org

0, 3, 0, 6, 2, 6, 16, 18, 22, 28, 15, 3, 5, 21, 46, 13, 58, 60, 33, 35, 8, 13, 41, 44, 96, 4, 34, 53, 108, 112, 42, 130, 8, 46, 148, 75, 78, 81, 166, 43, 178, 180, 95, 192, 98, 99, 30, 222, 113, 228, 232, 7, 30, 50, 256, 262, 268, 5, 69, 28, 141, 146, 153, 155, 312, 79, 110

Offset: 1

Lekraj Beedassy, May 28 2002

nonn

If prime(n) = p then a(n) is a divisor of p-1. - Amarnath Murthy, Nov 11 2002

			The 13th prime, 41, divides the repunit 11111, the smallest among all R(5k) which are multiples of 41.

Michael De Vlieger, Table of n, a(n) for n = 1..1000
P. De Geest, Repunits and Their Factors
Amarnath Murthy, On the divisors of Smarandache Unary Sequence, Smarandache Notions Journal, Vol. 11, 2000.
A. A. A. Steward, Factorization of Repunits[up to R(196)]. Cached copy, Sep 25 2000.
Michael De Vlieger, Records and positions in first 1000 terms of A071126

Number of 1's in A077573(n).

Cf. A000042. Apart from a(2), identical to A002371.

Table[Function[p, If[Divisible[10, p], 0, k = {1}; While[! Divisible[ FromDigits@ k, p], AppendTo[k, 1]]; Length@ k]]@ Prime@ n, {n, 67}] (* Michael De Vlieger, May 20 2017 *)

from sympy import prime
from itertools import count
def a(n):
    if n == 1 or n == 3: return 0
    pn = prime(n)
    return next(k for k in count(1) if 10**k//9%pn == 0)
print([a(n) for n in range(1, 68)]) # Michael S. Branicky, Jul 24 2025

A051627 Periods associated with A040017.

Original entry on oeis.org

1, 2, 3, 4, 10, 12, 9, 14, 24, 36, 48, 38, 19, 23, 39, 62, 120, 150, 106, 93, 134, 294, 196, 320, 654, 738, 385, 586, 317, 597, 1404, 945, 1452, 1836, 1752, 1172, 1812, 1282, 1426, 2232, 1862, 1844, 1521, 2134, 3750, 1031, 2264, 2667, 4354, 3927, 4274, 6522, 3903, 6022, 6682, 6135, 9550, 5877

Offset: 1

N. J. A. Sloane

The numbers in A007498 sorted according to the magnitude of the corresponding prime. - T. D. Noe, Sep 08 2005

			The decimal expansion of 1/101 is 0.00990099..., having a period of 4 and it is the only prime with that period.

Ray Chandler, Table of n, a(n) for n = 1..99
C. K. Caldwell, Unique Primes
Eric Weisstein's World of Mathematics, Unique Prime
Index entries for sequences related to decimal expansion of 1/n

nmax = 10000; primesPeriods = Reap[Do[p = Cyclotomic[n, 10]/GCD[n, Cyclotomic[n, 10]]; If[PrimeQ[p], Print[n]; Sow[{p, n}]], {n, 1, nmax}]][[2, 1]]; Sort[primesPeriods][[All, 2]] // Prepend[#, 1]& // Take[#, 58]& (* Jean-François Alcover, Mar 29 2013 *)

a(n) = A002371(A000720(A040017(n))). - Max Alekseyev, Oct 14 2022

More terms from Jud McCranie
More terms from T. D. Noe, Sep 08 2005
Corrected a(45)=3750 and extended by Ray Chandler, Oct 13 2008

A052983 Least multiple of n consisting of a succession of 1's followed by a succession of 0's.

Original entry on oeis.org

10, 10, 1110, 100, 10, 1110, 1111110, 1000, 1111111110, 10, 110, 11100, 1111110, 1111110, 1110, 10000, 11111111111111110, 1111111110, 1111111111111111110, 100, 1111110, 110, 11111111111111111111110, 111000, 100, 1111110, 1111111111111111111111111110

Offset: 1

Lekraj Beedassy, Jun 26 2003

All entries are differences of two terms of A000042. Since the pigeonhole principle guarantees that, for any m, two among the first m+1 entries of A000042 are congruent modulo m, their difference (i.e. belonging to this sequence) is therefore divisible by m, so that such numbers exist for all m. This sequence is thus infinite.

For n>1, a(n) consists of s 1's and t 0's, where s=A084681(X) and t is the greater of p or q (s=1 for X=1, t=1 for p=q=0), when we write n=X*Y with (X,Y)=1 and Y=2^p*5^q.

			We have a(6)=1110 because 6 divides 1110=6*185, the smallest such one with a string of 1's followed by that of 0's

Cf. A000042, A002371, A007732.

Cf. A084681.

f[n_] := Select[ Map[ FromDigits, IntegerDigits[ Table[ Sum[2^i, {i, k, j, -1}], {j, k, 1, -1}], 2]]/n, IntegerQ[ # ] & ]; g[n_] := Block[{k = 1}, While[ f[n] == {}, k++ ]; n*Min[ f[n]]]; Table[ g[n], {n, 1, 27}]
nn=30;With[{nos=Sort[Flatten[Table[FromDigits[Join[Table[1,{n}], Table[ 0,{i}]]],{n,nn},{i,5}]]]},Flatten[Table[Select[nos,Divisible[#,n]&,1],{n,nn}]]] (* Harvey P. Dale, Mar 09 2014 *)

a(n) = A276348(n) * n; A227362(a(n)) = 10. - Jaroslav Krizek, Aug 30 2016

Edited, corrected and extended by Robert G. Wilson v, Jun 26 2003

A087000 Half length of periodic part of decimal expansion of 1/p for those primes having a periodic part of even length.

Original entry on oeis.org

3, 1, 3, 8, 9, 11, 14, 23, 29, 30, 4, 22, 48, 2, 17, 54, 56, 21, 65, 4, 23, 74, 39, 83, 89, 90, 96, 49, 15, 111, 114, 116, 15, 25, 128, 131, 134, 14, 73, 156, 55, 168, 58, 16, 183, 93, 189, 191, 194, 100, 102, 209, 70, 216, 16, 76, 230, 77, 243, 245, 249, 251

Offset: 1

Reinhard Zumkeller, Jul 29 2003

a(n) appears to be the least k such that 10^k+1 is divisible by A028416(n). See A001271. - Michel Marcus, Aug 13 2023

Cf. A028416, A002371, A086999, A087001, A087002, A001271, A062397 (10^n+1).

a(n) = A002371(A049084(A028416(n)))/2.
a(n) = A055642(A086999(n))/2.
a(n) = A055642(A087001(n)) = A055642(A087002(n)).

A372801 Order of 16 modulo the n-th prime: least k such that prime(n) divides 16^k-1.

Original entry on oeis.org

1, 1, 3, 5, 3, 2, 9, 11, 7, 5, 9, 5, 7, 23, 13, 29, 15, 33, 35, 9, 39, 41, 11, 12, 25, 51, 53, 9, 7, 7, 65, 17, 69, 37, 15, 13, 81, 83, 43, 89, 45, 95, 24, 49, 99, 105, 37, 113, 19, 29, 119, 6, 25, 4, 131, 67, 135, 23, 35, 47, 73, 51, 155, 39, 79, 15, 21, 173, 87, 22, 179

Offset: 2

Jianing Song, May 13 2024

a(n) is the period of the expansion of 1/prime(n) in hexadecimal.

Jianing Song, Table of n, a(n) for n = 2..10000

Cf. A302141 (order of 16 mod 2n+1).

In other bases: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.

PARI
```
a(n) = znorder(Mod(16, prime(n))).
```

a(n) = A014664(n)/gcd(4, A014664(n)) = A082654(n)/gcd(2, A082654(n)).

a(n) <= (prime(n) - 1)/2.

A060259 Denoting 4 consecutive primes by p, q, r and s, these are the values of q such that q and r have 10 as a primitive root, but p and s do not.

Original entry on oeis.org

59, 109, 179, 229, 571, 701, 937, 1019, 1171, 1429, 1619, 1777, 1811, 1847, 2063, 2269, 2297, 2339, 2383, 2447, 2731, 2819, 2927, 3257, 3299, 3331, 3461, 3571, 3593, 3617, 3701, 3833, 3967, 4139, 4259, 4421, 4567, 4691, 4937, 5087, 5153, 5179, 5417

Offset: 1

Jeff Burch, Mar 23 2001

nonn

A prime p has 10 as a primitive root iff the length of the period of the decimal expansion of 1/p is p-1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001913, A002371, A060260, A060261, A060262.

test[p_] := MultiplicativeOrder[10, p]===p-1; Prime/@Select[Range[2, 800], test[Prime[ # ]]&&test[Prime[ #+1]]&&!test[Prime[ #-1]]&&!test[Prime[ #+2]]&]
Prime[#+1]&/@SequencePosition[Table[If[MultiplicativeOrder[10,p]===p-1,1,0],{p,Prime[Range[ 800]]}],{0,1,1,0}][[;;,1]] (* Harvey P. Dale, Nov 29 2023 *)

Edited by Dean Hickerson, Jun 17 2002

Offset corrected by Amiram Eldar, Oct 03 2021

A060260 Numbers k such that prime(k), prime(k+1) and prime(k+2) have 10 as a primitive root, but prime(k-1) and prime(k+3) do not.

Original entry on oeis.org

55, 75, 141, 164, 184, 199, 358, 371, 380, 432, 559, 702, 745, 808, 825, 858, 882, 1077, 1097, 1279, 1299, 1303, 1328, 1408, 1431, 1486, 1502, 1558, 1654, 1702, 1724, 1744, 1768, 1820, 1835, 1873, 1901, 1905, 1953, 1977, 2050, 2148, 2216, 2220, 2267

Offset: 1

Jeff Burch, Mar 23 2001

nonn

A prime p has 10 as a primitive root iff the length of the period of the decimal expansion of 1/p is p-1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The corresponding primes are in A060261.

Cf. A001913, A002371, A060259, A060262.

test[p_] := MultiplicativeOrder[10, p]===p-1; Select[Range[2, 2500], test[Prime[ # ]]&&test[Prime[ #+1]]&&test[Prime[ #+2]]&&!test[Prime[ #-1]]&&!test[Prime[ #+3]]&]

Edited by Dean Hickerson, Jun 17 2002

cp's OEIS Frontend

A385222 a(n) is the multiplicative order of -10 modulo prime(n); a(1) = a(3) = 0 for completion.

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A007450 Decimal expansion of 1/17.

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A007615 Primes with unique period length (the periods are given in A007498).

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A071126 Length of least repunit which is a multiple of the n-th prime, or 0 if no such multiple exists.

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A051627 Periods associated with A040017.

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A052983 Least multiple of n consisting of a succession of 1's followed by a succession of 0's.

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A087000 Half length of periodic part of decimal expansion of 1/p for those primes having a periodic part of even length.

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A372801 Order of 16 modulo the n-th prime: least k such that prime(n) divides 16^k-1.

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