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A002371
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Period of decimal expansion of 1/(n-th prime) (0 by convention for the primes 2 and 5).
(Formerly M4050 N1680)
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50
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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
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OFFSET
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1,4
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COMMENTS
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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
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REFERENCES
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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, p. 162.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 15.
W. 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.
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).
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LINKS
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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)
Eric Weisstein's World of Mathematics, Decimal Expansion
Index entries for sequences related to decimal expansion of 1/n
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FORMULA
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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)
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EXAMPLE
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A002371(11) = 15 because the 11th prime is 31, and 1/31 = 0.03225806451612903225806451612903225806452... has period 15. - Richard F. Lyon, Mar 29 2022
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MAPLE
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seq(subs(FAIL=0, numtheory:-order(10, ithprime(n))), n=1..100); # Robert Israel, Jul 15 2016
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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See A048595 for another version. Cf. A006883, A007732, A051626, A071126, A000040, A002275, A097443.
Cf. A001913 (full reptend primes), A060257 (1/prime(n) has period prime(n) - 1).
Sequence in context: A346835 A195474 A021945 * A048595 A302346 A340421
Adjacent sequences: A002368 A002369 A002370 * A002372 A002373 A002374
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KEYWORD
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nonn,nice,easy,base
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Arlin Anderson (starship1(AT)gmail.com)
Edited by Charles R Greathouse IV, Mar 24 2010
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STATUS
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approved
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