A007732 Period of decimal representation of 1/n.

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

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

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

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

A000732 := proc(n)
    a132740 := 1 ;
    for pe in ifactors(n)[2] do
        if not op(1, pe) in {2, 5} then
            a132740 := a132740*op(1, pe)^op(2, pe) ;
        end if;
    end do:
    if a132740 = 1 then
        1 ;
    else
        numtheory[order](10, a132740) ;
    end if;
end proc:
seq(A000732(n), n=1..50) ; # R. J. Mathar, May 05 2023

Mathematica

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

Prog

(PARI) a(n)=znorder(Mod(10, n/2^valuation(n, 2)/5^valuation(n, 5))) \\ Charles R Greathouse IV, Jan 14 2013
(Sage)
def a(n):
    n = ZZ(n)
    rad = 2**n.valuation(2) * 5**n.valuation(5)
    return Zmod(n // rad)(10).multiplicative_order()
[a(n) for n in range(1, 20)]
# F. Chapoton, May 03 2020
(Python)
from sympy import n_order, multiplicity
def A007732(n): return n_order(10, n//2**multiplicity(2, n)//5**multiplicity(5, n)) # Chai Wah Wu, Feb 07 2022

Crossrefs

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

Sequence in context: A324544 A323160 A323166 * A237835 A126795 A348929

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