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A170945 Least number k such that the decimal representation of 1/k has period Fibonacci(n). 1
3, 11, 27, 41, 73, 53, 43, 103, 1321, 497867, 323, 467, 11311, 20141, 12169, 232159532264041847249 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

The period of 1/k is the least integer p such that 10^p = 1 (mod k). The integer p is also known as the multiplicative order of 10 (mod k).

REFERENCES

Mohammad K. Azarian, The Generating Function for the Fibonacci Sequence, Missouri Journal of Mathematical Sciences, Vol. 2, No. 2, Spring 1990, pp. 78-79. Zentralblatt MATH, Zbl 1097.11516.

Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.

LINKS

Table of n, a(n) for n=2..17.

Dario Alejandro Alpern, Factorization using the Elliptic Curve Method

C. K. Caldwell, Fibonacci Numbers

EXAMPLE

p(k) is the period of 1/k, we obtain with k=3,11,27,41,73,53,43,103 p(3)=1,p(11)=2,p(27)=3,p(41)=5,p(73)=8, p(53)=13,p(43)=21, p(103)=34

MAPLE

For the great numbers (p > 70), the maple program is very slow. That's what we use an process of two steps: factoring 10^p-1 with elliptic curve method (see the first web site), and then, for each factor q(k), k=1, 2, ..., r computation the periods of 1/q(k) and keep the period q(i) such that q(i) = Fibonacci number. The 17th term required 3h 2m for the computing of (10^1597) -1 T:=array(0..100); U:=array(0..100); n0:=0:n1:=1:T[1] = 1:for i from 2 to 30 do: n2:=n0+n1:T[i]:=n2:n0:=n1:n1:=n2:od:U[1]:=3:U[2]:=3:for q from 3 to 10 do: p0:=T[q]: indic:=0:for n from 1 to 2000 do:for p from 1 to 150 while(irem(10^p, n)<>1 or gcd(n, 10)<>1 ) do:od: if irem(10^p, n) = 1 and gcd(n, 10) = 1 and p=p0 and indic=0 then U[q]:=n:indic:=1:else fi:od: od: for n from 1 to 10 do:print( U[n]):od:

CROSSREFS

Cf. A000045, A039834 (signed Fibonacci numbers), A002329, A072859 (periodic sequences), A003060

Sequence in context: A101612 A123928 A186301 * A164897 A212982 A164845

Adjacent sequences:  A170942 A170943 A170944 * A170946 A170947 A170948

KEYWORD

nonn,base

AUTHOR

Michel Lagneau, Feb 19 2010

EXTENSIONS

Edited by T. D. Noe, Apr 14 2010

STATUS

approved

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Last modified April 17 01:44 EDT 2021. Contains 343051 sequences. (Running on oeis4.)