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

Author

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

Keywords

• nonn
• base
• easy
• nice

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

a(n.n) >= a(n) where n.n is A020338(n). - Davide Rotondo, Jun 13 2024

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

Crossrefs

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

Programs

Maple

A007732 := 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(A007732(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 *)

PARI

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

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

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

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

Extensions

More terms from James Sellers, Feb 05 2000