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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]
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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.
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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FORMULA
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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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Cf. A001913 (full reptend primes), A060257 (1/prime(n) has period prime(n) - 1).
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KEYWORD
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nonn,nice,easy,base
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AUTHOR
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EXTENSIONS
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More terms from Arlin Anderson (starship1(AT)gmail.com)
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STATUS
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approved
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