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A007732 Period of decimal representation of 1/n. 31
1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 6, 6, 1, 1, 16, 1, 18, 1, 6, 2, 22, 1, 1, 6, 3, 6, 28, 1, 15, 1, 2, 16, 6, 1, 3, 18, 6, 1, 5, 6, 21, 2, 1, 22, 46, 1, 42, 1, 16, 6, 13, 3, 2, 6, 18, 28, 58, 1, 60, 15, 6, 1, 6, 2, 33, 16, 22, 6, 35, 1, 8, 3, 1, 18, 6, 6, 13, 1, 9, 5, 41, 6, 16, 21, 28, 2, 44, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Appears to be a divisor of A007733*A007736. - Henry Bottomley, Dec 20 2001

Primes p such that a(p)=p-1 are in A001913. [Dmitry Kamenetsky, Nov 13 2008]

When 1/n has a finite decimal expansion (namely, when n = 2^a*5^b), a(n) = 1 while A051626(n) = 0. - M. F. Hasler, Dec 14 2015

REFERENCES

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, pp. 159 etc.

LINKS

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

Project Euler, Reciprocal cycles: Problem 26

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

FORMULA

Note that if n=r*s where r is a power of 2 and s is odd then a(n)=a(s). Also if n=r*s where r is a power of 5 and s is not divisible by 5 then a(n) = a(s). So we just need a(n) for n not divisible by 2 or 5. This is the smallest number m such that n divides 10^m - 1; m is a divisor of phi(n), where phi = A000010.

phi(n) = n-1 only if n is prime and since a(n) divides phi(n), a(n) can only equal n-1 if n is prime. - Scott Hemphill (hemphill(AT)alumni.caltech.edu), Nov 23 2006

a(n)=a(A132740(n)); a(A132741(n))=a(A003592(n))=1. - Reinhard Zumkeller, Aug 27 2007

MAPLE

A007732 := proc(n)

    option remember;

    local npr, phid, m ;

    if n = 1 then

        1;

    else

        npr := n ;

        while (npr mod 2 )=0 do

            npr := npr/2 ;

        end do:

        while (npr mod 5 )=0 do

            npr := npr/5 ;

        end do:

        if npr <> n then

            return procname(npr) ;

        else

            phid := sort(convert(numtheory[divisors](numtheory[phi](npr)), list)) ;

            for m in phid do

                if (10^m-1) mod n = 0 then

                    return m;

                end if;

            end do;

        end if;

    end if;

end proc: # R. J. Mathar, Oct 17 2012

MATHEMATICA

Table[r = n/2^IntegerExponent[n, 2]/5^IntegerExponent[n, 5]; MultiplicativeOrder[10, r], {n, 100}] (* T. D. Noe, Oct 17 2012 *)

digitCycleLength[r_Rational, b_Integer?Positive] := MultiplicativeOrder[b, FixedPoint[ Quotient[#, GCD[#, b]] &, Denominator[r]]]; (* from Mathematica help file for MultiplicativeOrder *) Array[ digitCycleLength[1/#, 10] &, 100, 2] (* Robert G. Wilson v, Aug 18 2014 *)

PROG

(PARI) A007732(n, amax)={ if( n % 2== 0, return(A007732(n/2, amax)) ; ) ; if( n % 5== 0, return(A007732(n/5, amax)) ; ) ; for(m=1, amax, if( (10^m-1) % n == 0, return(m) ; ) ; ) ; return(-1) ; } { for(n=1, 100, print(n, " ", A007732(n, 500)) ; ) ; } \\ R. J. Mathar, Aug 30 2006

(PARI) a(n)=znorder(Mod(10, n/2^valuation(n, 2)/5^valuation(n, 5))) \\ Charles R Greathouse IV, Jan 14 2013

CROSSREFS

Cf. A121341, A066799, A121090, A001913, A084680.

Sequence in context: A080219 A040037 A009194 * A237835 A126795 A276997

Adjacent sequences:  A007729 A007730 A007731 * A007733 A007734 A007735

KEYWORD

nonn,base,easy,nice

AUTHOR

N. J. A. Sloane, Hal Sampson [ hals(AT)easynet.com ]

EXTENSIONS

More terms from James A. Sellers, Feb 05 2000

STATUS

approved

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Last modified June 29 01:52 EDT 2017. Contains 288857 sequences.