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A173491 a(n) is the least k such that the period of the decimal expansion of 1/k is A000204(n).

%I A173491 #21 Feb 17 2019 20:24:45
%S A173491 3,27,101,239,21649,19,3191,35121409,722817036322379041,2241,797,967,
%T A173491 1230610745978027,3373,60787,509538919,15060275578609,5779,37397,
%U A173491 423557
%N A173491 a(n) is the least k such that the period of the decimal expansion of 1/k is A000204(n).
%C A173491 Smallest k such that A007732(k) = A000204(n).
%C A173491 For the large numbers (p > 70), the Maple program below is very slow. So we use a process of two steps: first, factor 10^p-1 using the elliptic curve method; then, for each factor q(k), k=1,2,...,r, compute the period of 1/q(k) and keep the period q(i) such that q(i) ... [unfinished sentence? - _R. J. Mathar_, Feb 24 2010] Compare the Maple section of A170945!
%D A173491 V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.
%D A173491 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
%D A173491 S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.
%H A173491 R. Javonovic, <a href="http://milan.milanovic.org/math/english/function1/function1.html">Lucas Function Calculator</a>
%e A173491 a(1)=3 because the period of 1/3 = 0.333... is 1, and 3 is the smallest number with that period.
%e A173491 a(2)=27 because the period of 1/27 = 0.037037... is 3 = A000204(2), and 27 is the smallest number with that period.
%e A173491 a(3)=101 because the period of 1/101 = 0.00990099... is 4 = A000204(3), and 101 is the smallest number with that period.
%e A173491 a(4)= 239 because the period of 1/239 = 0.00418410041841... is 7 = A000204(4), and 239 is the smallest number with that period.
%p A173491 T:=array(0..100); U:=array(0..100); n0:=1: n1:=3: T[1] := 1: T[2] := 3:for i from 3 to 30 do: n2:=n0+n1: T[i]:=n2: n0:=n1: n1:=n2: od:
%p A173491 for q from 1 to 7 do: p0:=T[q]: indic:=0: for n from 1 to 25000 do: for p from 1 to 30 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:
%p A173491 for n from 1 to 7 do: print( U[n]): od:
%t A173491 (* This [slow] mma program gives all denominators < 50000 and disagrees with existing sequence for n = 11: a(11) = 797 instead of 29453 *) a204[n_] := a204[n] = Coefficient[Series[(2 - t )/(1 - t - t^2), {t, 0, n}], t^n] ; a7732[n_] := a7732[n] = MultiplicativeOrder[10, FixedPoint[Quotient[#, GCD[#, 10]] &, n]]; a[n_] := (k = 2; While[k++; k < 50000 && a7732[k] != a204[n] ]; k); Table[a[n], {n, 1, 15}](* _Jean-François Alcover_, Sep 02 2011 *)
%Y A173491 Cf. A000032, A000204, A002329, A007732, A051626, A064362, A072859, A170945.
%K A173491 nonn,base
%O A173491 1,1
%A A173491 _Michel Lagneau_, Feb 19 2010
%E A173491 References to unrelated sequences removed by _R. J. Mathar_, Feb 24 2010
%E A173491 Extended with the help of _Jean-François Alcover_ and _D. S. McNeil_ by _T. D. Noe_, Sep 07 2011