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A036275 The periodic part of the decimal expansion of 1/n. Any initial 0's are to be placed at end of cycle.

%I A036275 #63 Feb 12 2021 18:15:57
%S A036275 0,0,3,0,0,6,142857,0,1,0,90,3,769230,714285,6,0,5882352941176470,5,
%T A036275 526315789473684210,0,476190,45,4347826086956521739130,6,0,384615,370,
%U A036275 571428,3448275862068965517241379310,3,322580645161290,0,30,2941176470588235,285714,7
%N A036275 The periodic part of the decimal expansion of 1/n. Any initial 0's are to be placed at end of cycle.
%C A036275 a(n) = 0 iff n = 2^i*5^j (A003592). - _Jon Perry_, Nov 19 2014
%C A036275 a(n) = n iff n = 3 or 6 (see De Koninck & Mercier reference). - _Bernard Schott_, Dec 02 2020
%D A036275 Jean-Marie De Koninck & Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 347 pp. 50 and 205, Ellipses, Paris, 2004.
%H A036275 Philippe Guglielmetti, <a href="/A036275/b036275.txt">Table of n, a(n) for n = 1..1001</a> (first 500 terms from T. D. Noe)
%H A036275 <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>
%e A036275 1/28 = .03571428571428571428571428571428571428571... and digit-cycle is 571428, so a(28)=571428.
%p A036275 isCycl := proc(n) local ifa,i ; if n <= 2 then RETURN(false) ; fi ; ifa := ifactors(n)[2] ; for i from 1 to nops(ifa) do if op(1,op(i,ifa)) <> 2 and op(1,op(i,ifa)) <> 5 then RETURN(true) ; fi ; od ; RETURN(false) ; end: A036275 := proc(n) local ifa,sh,lpow,mpow,r ; if not isCycl(n) then RETURN(0) ; else lpow:=1 ; while true do for mpow from lpow-1 to 0 by -1 do if (10^lpow-10^mpow) mod n =0 then r := (10^lpow-10^mpow)/n ; r := r mod (10^(lpow-mpow)-1) ; while r*10 < 10^(lpow-mpow) do r := 10*r ; od ; RETURN(r) ; fi ; od ; lpow := lpow+1 ; od ; fi ; end: for n from 1 to 60 do printf("%d %d ",n,A036275(n)) ; od ; # _R. J. Mathar_, Oct 19 2006
%t A036275 fc[n_]:=Block[{q=RealDigits[1/n][[1,-1]]},If[IntegerQ[q],0,While[First[q]==0,q=RotateLeft[q]];FromDigits[q]]];
%t A036275 Table[fc[n],{n,36}] (* _Ray Chandler_, Nov 19 2014, corrected Jun 27 2017 *)
%t A036275 Table[FromDigits[FindTransientRepeat[RealDigits[1/n,10,120][[1]],3] [[2]]],{n,40}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Mar 12 2019 *)
%Y A036275 Cf. A007732, A051626, A002371, A048595, A006883, A007498, A007615, A040017, A051627.
%Y A036275 See also A060282, A060283, A060251.
%Y A036275 A051628 is length of preamble.
%K A036275 base,nonn,easy,nice
%O A036275 1,3
%A A036275 _Floor van Lamoen_
%E A036275 Corrected and extended by _N. J. A. Sloane_
%E A036275 Corrected a(92), a(208), a(248), a(328), a(352) and a(488) which missed a trailing zero (see the table). - _Philippe Guglielmetti_, Jun 20 2017