

A007732


Period of decimal representation of 1/n.


36



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
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OFFSET

1,7


COMMENTS

Appears to be a divisor of A007733*A007736.  Henry Bottomley, Dec 20 2001
Primes p such that a(p)=p1 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) = n1 only if n is prime and since a(n) divides phi(n), a(n) can only equal n1 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


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


CROSSREFS

Cf. A121341, A066799, A121090, A001913, A084680.
Sequence in context: A324544 A323160 A323166 * A237835 A126795 A334491
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



