A071126 -id:A071126 - cp's OEIS frontend

A002371 Period of decimal expansion of 1/(n-th prime) (0 by convention for the primes 2 and 5).

0, 1, 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

N. J. A. Sloane

a(n) is the minimum solution x of modular equation 10^x == 1 (mod p), where p = prime(n). - Carmine Suriano, Oct 10 2012
a(n) = smallest m such that 111...11 (m 1's) is divisible by the n-th prime, or 0 if no such m exists (with the exception that a(2) = 3 instead of 1). E.g., the 5th prime, 11, divides 11, so a(5) = 2. - N. J. A. Sloane, Oct 03 2013 [Comment corrected by Derek Orr, Jun 14 2014]
Numbers n such that A071126(n) = A000040(n) - 1. - Hugo Pfoertner, Mar 18 2003
Except for n = 1 and 3, a(n) divides A006093(n). - Robert Israel, Jul 15 2016

			A002371(11) = 15 because the 11th prime is 31, and 1/31 = 0.03225806451612903225806451612903225806452... has period 15. - _Richard F. Lyon_, Mar 29 2022

Albert H. Beiler, Recreations in the Theory of Numbers, 2nd ed. New York: Dover, 1966, pages 65, 309. ISBN 0-486-21096-0.
John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, 1996, p. 162. ISBN 978-0-387-97993-9.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 15.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
C. K. Caldwell, The Prime Glossary, Period of a prime
Matt Parker and Brady Haran, The Reciprocals of Primes, Numberphile video (2022).
William Shanks, On the number of figures in the period of the reciprocal of every prime number below 20,000, Proc. Royal Soc. London, 22 (1874), 200-210. See also on JSTOR.
Eric Weisstein's World of Mathematics, Decimal Expansion
Index entries for sequences related to decimal expansion of 1/n

See A048595 for another version. Cf. A006883, A007732, A051626, A071126, A000040, A002275, A097443.
Cf. A001913 (full repetend primes), A060257 (1/prime(n) has period prime(n) - 1).
In other bases: A014664, A062117, A211241, A211242, A211243, A211244, A211245, A002371.

seq(subs(FAIL=0,numtheory:-order(10, ithprime(n))),n=1..100); # Robert Israel, Jul 15 2016

Table[ Length[ RealDigits[1 / Prime[n]] [[1, 1]]], {n, 1, 70}]
Table[If[IntegerQ[#], #, 0] &[MultiplicativeOrder[10, Prime[n]]], {n, 1, 70}] (* Jan Mangaldan, Jul 07 2020 *)

a(n)=if(n<4,n==2,znorder(Mod(10, prime(n))))

from sympy import prime, n_order
def A002371(n): return 0 if n == 1 or n == 3 else n_order(10,prime(n)) # Chai Wah Wu, Feb 07 2022

From Alexander Adamchuk, Jan 28 2007: (Start)
a(A000720(p)) = p - 1 for primes p in A001913.
a(A060257(n)) = prime(A060257(n)) - 1. (End)

More terms from Arlin Anderson (starship1(AT)gmail.com)

Edited by Charles R Greathouse IV, Mar 24 2010

A001914 Cyclic numbers: 10 is a quadratic residue modulo p and class of mantissa is 2.

Original entry on oeis.org

2, 13, 31, 43, 67, 71, 83, 89, 107, 151, 157, 163, 191, 197, 199, 227, 283, 293, 307, 311, 347, 359, 373, 401, 409, 431, 439, 443, 467, 479, 523, 557, 563, 569, 587, 599, 601, 631, 653, 677, 683, 719, 761, 787, 827, 839, 877, 881, 883, 911, 919, 929, 947, 991

Offset: 1

N. J. A. Sloane

nonn

Also, apart from first term 2, primes p for which the repunit (A002275) R((p-1)/2)=(10^((p-1)/2)-1)/9 is the smallest repunit divisible by p. Primes for which A000040(n) = 2*A071126(n) + 1. - Hugo Pfoertner, Mar 18 2003, Sep 18 2018

			The repunit R(6)=111111 is the smallest repunit divisible by the prime a(2)=13=2*6+1.

Albert H. Beiler, Recreations in the Theory of Numbers, 2nd ed. New York: Dover, 1966. Pages 65, 309.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 61.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Hugo Pfoertner, Table of n, a(n) for n = 1..1180

Cf. A003277 for another sequence of cyclic numbers.

Cf. A000040, A002275, A071126.

R(n)=(10^n-1)/9;
print1(2,", "); forprime(p=3, 1000, m=0; for(q=3, (p-1)/2-1, if(R(q)%p==0, m=1; break));if(m==0&&R((p-1)/2)%p==0, print1(p,", "))) \\ Hugo Pfoertner, Sep 18 2018

More terms from Enoch Haga

A077573 Smallest number of the form (10^k -1)/9 == 0 (mod prime(n)). with a(1) = a(3) = 0.

Original entry on oeis.org

0, 111, 0, 111111, 11, 111111, 1111111111111111, 111111111111111111, 1111111111111111111111, 1111111111111111111111111111, 111111111111111, 111, 11111, 111111111111111111111, 1111111111111111111111111111111111111111111111

Offset: 1

Amarnath Murthy, Nov 11 2002

nonn

For every prime p > 5, {10^(p-1) -1}/9 == 0 (mod p), by Fermat's Little theorem.

Amarnath Murthy, On the divisors of Smarandache Unary Sequence. Smarandache Notions Journal, Vol. 11, 2000.

Alois P. Heinz, Table of n, a(n) for n = 1..170

Cf. A071126.

More terms from Sascha Kurz, Jan 04 2003

A212528 The periodic part of the decimal expansion of prime(n-1) / prime(n).

Original entry on oeis.org

6, 0, 714285, 63, 846153, 7647058823529411, 894736842105263157, 8260869565217391304347, 7931034482758620689655172413, 935483870967741, 837, 90243, 953488372093023255813, 9148936170212765957446808510638297872340425531, 8867924528301

Offset: 2

Jaroslav Krizek, Jun 29 2012

Number of digits of the periodic parts for n>=3 in A002371, A048595 and A071126.

Jaroslav Krizek, Table of n, a(n) for n = 2..60

Cf. A002371, A048595, A071126.

Table[If[n == 3, 0, FromDigits[RealDigits[Prime[n-1]/Prime[n]][[1,1]]]], {n, 2, 10}]

Corrected by T. D. Noe, Jun 29 2012

cp's OEIS Frontend

A077574 Duplicate of A071126.

Views

Author

Keywords

A002371 Period of decimal expansion of 1/(n-th prime) (0 by convention for the primes 2 and 5).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A001914 Cyclic numbers: 10 is a quadratic residue modulo p and class of mantissa is 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Extensions

A077573 Smallest number of the form (10^k -1)/9 == 0 (mod prime(n)). with a(1) = a(3) = 0.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A212528 The periodic part of the decimal expansion of prime(n-1) / prime(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A077575 a(n) = A077573(n)/prime(n).

Views

Author

Keywords

Comments

References

Crossrefs

Extensions