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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy
• base

Comments

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

Examples

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

References

• 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).

Links

• 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

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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)

Extensions

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

Edited by Charles R Greathouse IV, Mar 24 2010