A002371 -id:A002371 - cp's OEIS frontend

A036275 The periodic part of the decimal expansion of 1/n. Any initial 0's are to be placed at end of cycle.

0, 0, 3, 0, 0, 6, 142857, 0, 1, 0, 90, 3, 769230, 714285, 6, 0, 5882352941176470, 5, 526315789473684210, 0, 476190, 45, 4347826086956521739130, 6, 0, 384615, 370, 571428, 3448275862068965517241379310, 3, 322580645161290, 0, 30, 2941176470588235, 285714, 7

Offset: 1

Floor van Lamoen

a(n) = 0 iff n = 2^i*5^j (A003592). - Jon Perry, Nov 19 2014

a(n) = n iff n = 3 or 6 (see De Koninck & Mercier reference). - Bernard Schott, Dec 02 2020

			1/28 = .03571428571428571428571428571428571428571... and digit-cycle is 571428, so a(28)=571428.

Jean-Marie De Koninck & Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 347 pp. 50 and 205, Ellipses, Paris, 2004.

Philippe Guglielmetti, Table of n, a(n) for n = 1..1001 (first 500 terms from T. D. Noe)
Index entries for sequences related to decimal expansion of 1/n

Cf. A007732, A051626, A002371, A048595, A006883, A007498, A007615, A040017, A051627.
See also A060282, A060283, A060251.
A051628 is length of preamble.

isCycl := proc(n) local ifa,i ; if n <= 2 then RETURN(false) ; fi ; ifa := ifactors(n)[2] ; for i from 1 to nops(ifa) do if op(1,op(i,ifa)) <> 2 and op(1,op(i,ifa)) <> 5 then RETURN(true) ; fi ; od ; RETURN(false) ; end: A036275 := proc(n) local ifa,sh,lpow,mpow,r ; if not isCycl(n) then RETURN(0) ; else lpow:=1 ; while true do for mpow from lpow-1 to 0 by -1 do if (10^lpow-10^mpow) mod n =0 then r := (10^lpow-10^mpow)/n ; r := r mod (10^(lpow-mpow)-1) ; while r*10 < 10^(lpow-mpow) do r := 10*r ; od ; RETURN(r) ; fi ; od ; lpow := lpow+1 ; od ; fi ; end: for n from 1 to 60 do printf("%d %d ",n,A036275(n)) ; od ; # R. J. Mathar, Oct 19 2006

fc[n_]:=Block[{q=RealDigits[1/n][[1,-1]]},If[IntegerQ[q],0,While[First[q]==0,q=RotateLeft[q]];FromDigits[q]]];
Table[fc[n],{n,36}] (* Ray Chandler, Nov 19 2014, corrected Jun 27 2017 *)
Table[FromDigits[FindTransientRepeat[RealDigits[1/n,10,120][[1]],3] [[2]]],{n,40}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 12 2019 *)

Corrected and extended by N. J. A. Sloane

Corrected a(92), a(208), a(248), a(328), a(352) and a(488) which missed a trailing zero (see the table). - Philippe Guglielmetti, Jun 20 2017

A211242 Order of 6 mod n-th prime: least k such that prime(n) divides 6^k-1.

Original entry on oeis.org

0, 0, 1, 2, 10, 12, 16, 9, 11, 14, 6, 4, 40, 3, 23, 26, 58, 60, 33, 35, 36, 78, 82, 88, 12, 10, 102, 106, 108, 112, 126, 130, 136, 23, 37, 150, 156, 27, 83, 43, 178, 60, 19, 96, 14, 198, 105, 222, 226, 228, 232, 17, 20, 250, 256, 131, 134, 270, 276, 56, 141

Offset: 1

T. D. Noe, Apr 11 2012

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A019336 (full reptend primes in base 6).

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

A000040:=Filtered([1..350],IsPrime);;
List([1..Length(A000040)],n->OrderMod(6,A000040[n])); # Muniru A Asiru, Feb 06 2019

A211242 := proc(n)
    if n<= 2 then
        0 ;
    else
        numtheory[order](6,ithprime(n)) ;
    end if;
end proc:
seq(A211242(n),n=1..80) ; # R. J. Mathar, Jul 17 2024

nn = 6; Table[If[Mod[nn, p] == 0, 0, MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

a(n,{base=6}) = my(p=prime(n)); if(base%p, znorder(Mod(base,p)), 0) \\ Jianing Song, May 13 2024

A211245 Order of 9 mod n-th prime: least k such that prime(n) divides 9^k-1.

Original entry on oeis.org

1, 0, 2, 3, 5, 3, 8, 9, 11, 14, 15, 9, 4, 21, 23, 26, 29, 5, 11, 35, 6, 39, 41, 44, 24, 50, 17, 53, 27, 56, 63, 65, 68, 69, 74, 25, 39, 81, 83, 86, 89, 45, 95, 8, 98, 99, 105, 111, 113, 57, 116, 119, 60, 125, 128, 131, 134, 15, 69, 140, 141, 146, 17, 155, 39

Offset: 1

T. D. Noe, Apr 11 2012

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A364867.

In other bases: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A002371, A372801.

A000040:=Filtered([1..350],IsPrime);;
List([1..Length(A000040)],n->OrderMod(9,A000040[n])); # Muniru A Asiru, Feb 06 2019

nn = 9; Table[If[Mod[nn, p] == 0, 0, MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

a(n,{base=9}) = my(p=prime(n)); if(base%p, znorder(Mod(base,p)), 0) \\ Jianing Song, May 13 2024

From Jianing Song, May 13 2024: (Start)
a(n) = A062117(n)/gcd(2, A062117(n)).
a(n) <= (prime(n) - 1)/2. Those prime(n) for which a(n) = (prime(n) - 1)/2 are listed in A364867. (End)

A211243 Order of 7 mod n-th prime: least k such that prime(n) divides 7^k-1.

Original entry on oeis.org

1, 1, 4, 0, 10, 12, 16, 3, 22, 7, 15, 9, 40, 6, 23, 26, 29, 60, 66, 70, 24, 78, 41, 88, 96, 100, 51, 106, 27, 14, 126, 65, 68, 69, 74, 150, 52, 162, 83, 172, 178, 12, 10, 24, 98, 99, 210, 37, 113, 228, 116, 238, 240, 125, 256, 262, 268, 135, 138, 20, 141, 292

Offset: 1

T. D. Noe, Apr 11 2012

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A019337 (full reptend primes in base 7).
In other bases: A014664, A062117, A082654, A211241, A211242, A211244, A211245, A002371.
Row lengths of A201911. - Michel Marcus, Feb 04 2019

A000040:=Filtered([1..350],IsPrime);;
List([1..Length(A000040)],n->OrderMod(7,A000040[n])); # Muniru A Asiru, Feb 06 2019

nn = 7; Table[If[Mod[nn, p] == 0, 0, MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

a(n,{base=7}) = my(p=prime(n)); if(base%p, znorder(Mod(base,p)), 0) \\ Jianing Song, May 13 2024

A211244 Order of 8 mod n-th prime: least k such that prime(n) divides 8^k-1.

Original entry on oeis.org

0, 2, 4, 1, 10, 4, 8, 6, 11, 28, 5, 12, 20, 14, 23, 52, 58, 20, 22, 35, 3, 13, 82, 11, 16, 100, 17, 106, 12, 28, 7, 130, 68, 46, 148, 5, 52, 54, 83, 172, 178, 60, 95, 32, 196, 33, 70, 37, 226, 76, 29, 119, 8, 50, 16, 131, 268, 45, 92, 70, 94, 292, 34, 155, 52

Offset: 1

T. D. Noe, Apr 11 2012

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A053451 (order of 8 mod 2n+1), A019338 (full reptend primes in base 8).

In other bases: A014664, A062117, A082654, A211241, A211242, A211243, A211245, A002371, A372801.

A000040:=Filtered([1..350],IsPrime);;
List([1..Length(A000040)],n->OrderMod(8,A000040[n])); # Muniru A Asiru, Feb 06 2019

nn = 8; Table[If[Mod[nn, p] == 0, 0, MultiplicativeOrder[nn, p]], {p, Prime[Range[100]]}]

a(n,{base=8}) = my(p=prime(n)); if(base%p, znorder(Mod(base,p)), 0) \\ Jianing Song, May 13 2024

a(n) = A014664(n)/gcd(3, A014664(n)). - Jianing Song, May 13 2024

A040017 Prime 3 followed by unique period primes (the period r of 1/p is not shared with any other prime) of the form A019328(r)/gcd(A019328(r),r) in order (periods r are given in A051627).

Original entry on oeis.org

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

Offset: 1

Jud McCranie

Prime p=3 is the only known example of a unique period prime such that A019328(r)/gcd(A019328(r),r) = p^k with k > 1 (cf. A323748). It is plausible to assume that no other such prime exists. Under this (unproved) assumption, the current sequence lists all unique period primes in order and represents a sorted version of A007615. - Max Alekseyev, Oct 14 2022

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

J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), p. 324, Pour la Science Paris 2000.

Robert G. Wilson v, Table of n, a(n) for n = 1..47
Chris Caldwell, The Prime Glossary, Unique prime.
C. K. Caldwell, "Top Twenty" page, Unique.
Chris K. Caldwell and Harvey Dubner, Unique-Period Primes, J. Recreational Math., 29:1 (1998) 43-48.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Eric Weisstein's World of Mathematics, Unique Prime.
Wikipedia, Unique prime.
Index entries for sequences related to decimal expansion of 1/n

Cf. A007615, A007498, A002371, A048595, A006883, A007732, A051626, A051627, A323748.

lst = {}; Do[c = Cyclotomic[n, 10]; q = c/GCD[c, n]; If[PrimeQ[q], AppendTo[lst, q]], {n, 62}]; Prepend[Sort[lst], 3] (* Arkadiusz Wesolowski, May 13 2012 *)

For n >= 2, a(n) = A019328(r) / gcd(A019328(r), r), where r = A051627(n). - Max Alekseyev, Oct 14 2022

Missing term a(45) inserted in b-file at the suggestion of Eric Chen by Max Alekseyev, Oct 13 2022

Edited by Max Alekseyev, Oct 14 2022

A028416 Primes p such that the decimal expansion of 1/p has a periodic part of even length.

Original entry on oeis.org

7, 11, 13, 17, 19, 23, 29, 47, 59, 61, 73, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 193, 197, 211, 223, 229, 233, 241, 251, 257, 263, 269, 281, 293, 313, 331, 337, 349, 353, 367, 373, 379, 383, 389, 401, 409, 419, 421, 433

Offset: 1

Mario Velucchi (mathchess(AT)velucchi.it)

Primes whose reciprocals have even period length.
Primes p such that the order of 10 mod p is even. - Joerg Arndt, Mar 04 2014
A002371(A049084(a(n))) mod 2 == 0.
Not the same as A040121: a(33)=241 is not in A040121.
Let (d(i): 1<=i<=2*K) be the period of the decimal expansion of 1/a(n), K=A002371(A049084(a(n)))/2, then d(i) + d(i+K) = 9 for i with 1<=i<=K, or, equivalently: u + v = 10^K - 1 with u = Sum_{i=1..K} d(i)*10^(K-i) and v = Sum_{i=1..K} d(i+K)*10^(K-i). - Reinhard Zumkeller, Oct 05 2008

			From _Reinhard Zumkeller_, Oct 05 2008: (Start)
(0,5,8,8,2,3,5,2,9,4,1,1,7,6,4,7) is the period of 1/17 (see A007450),
K = A002371(A049084(17))/2 = A002371(7)/2 = 16/2 = 8,
u = 5882352, v = 94117647: u + v = 99999999 = 10^8 - 1. (End)

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, "Die periodischen Dezimalbrueche". [Reinhard Zumkeller, Oct 05 2008]

Cf. A087000, A186635.

A028416 := proc(n) local st:
st := ithprime(n):
if (modp(numtheory[order](10,st),2) = 0) then
   RETURN(st)
fi: end:  seq(A028416(n), n=1..100); # Jani Melik, Feb 24 2011

Select[Prime[Range[4,100]],EvenQ[Length[RealDigits[1/#][[1,1]]]]&] (* Harvey P. Dale, Jul 07 2011 *)

forprime(p=7,1e3,if(znorder(Mod(10,p))%2==0,print1(p", "))) \\ Charles R Greathouse IV, Feb 24 2011

from sympy import gcd, isprime, n_order
is_A028416 = lambda n: gcd(n,10)==1 and n>5 and n_order(10, n)%2==0 and isprime(n) # M. F. Hasler, Nov 19 2024

More terms from Reinhard Zumkeller, Jul 29 2003

A056210 Primes p whose period of reciprocal equals (p-1)/5.

Original entry on oeis.org

11, 251, 1061, 1451, 1901, 1931, 2381, 3181, 3491, 3851, 4621, 4861, 5261, 6101, 6491, 6581, 6781, 7331, 8101, 9941, 10331, 10771, 11251, 11261, 11411, 12301, 14051, 14221, 14411, 15091, 15131, 16061, 16141, 16301, 16651, 16811, 16901

Offset: 1

Robert G. Wilson v, Aug 02 2000

Cyclic numbers of the fifth degree (or fifth order): the reciprocals of these numbers belong to one of five different cycles. Each cycle has the (number minus 1)/5 digits.
From Robert Israel, Apr 02 2018: (Start)
Primes p such that A002371(A000720(p)) = (p-1)/5.
All terms == 1 (mod 10). (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Index entries for sequences related to decimal expansion of 1/n

Cf. A000720, A001913, A002371, A097443, A055628, A056157, A056211, A056212, A056213, A056214, A056215, A056216, A056217, A098680.

select(t -> isprime(t) and numtheory:-order(10, t) = (t-1)/5, [seq(t,t=11..17000,10)]); # Robert Israel, Apr 02 2018

f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 2000]], f[ # ] == 5 &]

Entry revised by N. J. A. Sloane, Apr 30 2007

A056213 Primes p for which the period of reciprocal = (p-1)/8.

Original entry on oeis.org

41, 241, 1601, 1609, 2441, 2969, 3041, 3449, 3929, 4001, 4409, 5009, 6089, 6521, 6841, 8161, 8329, 8609, 9001, 9041, 9929, 13001, 13241, 14081, 14929, 16001, 16481, 17489, 17881, 18121, 19001, 20249, 20641, 20921, 21529, 22481, 23801

Offset: 1

Robert G. Wilson v, Aug 02 2000

Cyclic numbers of the eighth degree (or eighth order): the reciprocals of these numbers belong to one of eight different cycles. Each cycle has the (number minus 1)/8 digits.
From Robert Israel, Apr 02 2018: (Start)
Primes p such that A002371(A000720(p))=(p-1)/8.
All terms == 1 (mod 8). (End)

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of 1/n

select(t -> isprime(t) and numtheory:-order(10, t) = (t-1)/8, [seq(t,t=17..24000,8)]); # Robert Israel, Apr 02 2018

f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 2700]], f[ # ] == 8 &]

Edited by N. J. A. Sloane, Apr 30 2007

A007498 Unique period lengths of primes mentioned in A007615.

Original entry on oeis.org

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

Offset: 1

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

Let {Zs(m, 10, 1)} be the Zsigmondy numbers for a = 10, b = 1: Zs(m, 10, 1) is the greatest divisor of 10^m - 1^m that is coprime to 10^r - 1^r for all positive integers r < m. Then this sequence gives m such that Zs(m, 10, 1) is a prime power (e.g., Zs(1, 10, 1) = 9 = 3^2, Zs(2, 10, 1) = 11, Zs(3, 10, 1) = 37, Zs(4, 10, 1) = 101). It is very likely that Zs(m, 10, 1) is prime if m > 1 is in this sequence (note that the Mathematica and PARI programs below are based on this assumption). - Jianing Song, Aug 12 2020

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.

Ray Chandler, Table of n, a(n) for n = 1..106
Chris K. Caldwell, Unique (period) primes and the factorization of cyclotomic polynomials minus one, Mathematica Japonica, 26 (1997), 189-195.
Chris K. Caldwell & H. Dubner, Unique-Period Primes, Table 2 in Journal of Recreational Mathematics 29(1) 46 1998.
Robert G. Wilson v, Notes, n.d.
Index entries for sequences related to decimal expansion of 1/n

Cf. A007615, A002371, A048595, A006883, A007732, A051626, A046108.

Cf. A161508 (unique period lengths in base 2).

lst={1}; Do[p=Cyclotomic[n, 10]/GCD[n, Cyclotomic[n, 10]]; If[PrimeQ[p], AppendTo[lst, n]], {n, 3000}]; lst (* T. D. Noe, Sep 08 2005 *)

isok(n) = if (n==1, 1, my(p = polcyclo(n, 10)); isprime(p/gcd(p, n))); \\ Michel Marcus, Jun 20 2018

More terms from T. D. Noe, Sep 08 2005

a(48)-a(52) from Ray Chandler, Jul 09 2008

cp's OEIS Frontend

A036275 The periodic part of the decimal expansion of 1/n. Any initial 0's are to be placed at end of cycle.

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A211242 Order of 6 mod n-th prime: least k such that prime(n) divides 6^k-1.

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A211245 Order of 9 mod n-th prime: least k such that prime(n) divides 9^k-1.

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A211243 Order of 7 mod n-th prime: least k such that prime(n) divides 7^k-1.

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A211244 Order of 8 mod n-th prime: least k such that prime(n) divides 8^k-1.

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A040017 Prime 3 followed by unique period primes (the period r of 1/p is not shared with any other prime) of the form A019328(r)/gcd(A019328(r),r) in order (periods r are given in A051627).

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A028416 Primes p such that the decimal expansion of 1/p has a periodic part of even length.

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