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A002371 Period of decimal expansion of 1/(n-th prime) (0 by convention for the primes 2 and 5).
(Formerly M4050 N1680)
45
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

a( PrimePi[p] ) = p - 1 for prime p = {7, 17, 19, 23, 29, 47, 59, 61, 97, ...} = A001913(n) Cyclic numbers: primes with primitive root 10. a( A060257(n) ) = prime( A060257(n) ) - 1, where A060257(n) = {4, 7, 8, 9, 10, 15, 17, 18, 25, 29, 30, 32, ...}. Numbers n such that 1/prime(n) has period prime(n) - 1. - Alexander Adamchuk, Jan 28 2007

Except for n=1 and 3, a(n) divides A006093(n). - Robert Israel, Jul 15 2016

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, 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).

LINKS

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

C. K. Caldwell, The Prime Glossary, Period of a prime

Eric Weisstein's World of Mathematics, Decimal Expansion

Index entries for sequences related to decimal expansion of 1/n

FORMULA

a(n) = x is the minimum solution of modular equation 10^x = 1 (mod p), where p = prime(n). - Carmine Suriano, Oct 10 2012

EXAMPLE

1/31 = .03225806451612903225806451612903225806452... has period 15.

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}]

PROG

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

CROSSREFS

See A048595 for another version. Cf. A006883, A007732, A051626, A071126, A000040, A002275.

Cf. A001913 = Cyclic numbers: primes with primitive root 10. Cf. A060257 = numbers n such that 1/prime(n) has period prime(n) - 1.

Cf. A275081.

Sequence in context: A195474 A021945 * A048595 A244922 A153313 A096050

Adjacent sequences:  A002368 A002369 A002370 * A002372 A002373 A002374

KEYWORD

nonn,nice,easy,base

AUTHOR

N. J. A. Sloane, Apr 30 1991

EXTENSIONS

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

Edited by Charles R Greathouse IV, Mar 24 2010

STATUS

approved

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Last modified August 17 13:03 EDT 2017. Contains 290635 sequences.