A036275 - OEIS

A036275

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

0, 0, 3, 0, 0, 6, 142857, 0, 1, 0, 90, 3, 769230, 714285, 6, 0, 5882352941176470, 5, 526315789473684210, 0, 476190, 45, 4347826086956521739130, 6, 0, 384615, 370, 571428, 3448275862068965517241379310, 3, 322580645161290, 0, 30, 2941176470588235, 285714, 7

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OFFSET

1,3

COMMENTS

a(n) = 0 iff n = 2^i*5^j (A003592). - Jon Perry, Nov 19 2014

a(n) = n iff n = 3 or 6 (see De Koninck & Mercier reference). - Bernard Schott, Dec 02 2020

REFERENCES

Jean-Marie De Koninck & Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 347 pp. 50 and 205, Ellipses, Paris, 2004.

LINKS

Philippe Guglielmetti, Table of n, a(n) for n = 1..1001 (first 500 terms from T. D. Noe)

Index entries for sequences related to decimal expansion of 1/n

EXAMPLE

1/28 = .03571428571428571428571428571428571428571... and digit-cycle is 571428, so a(28)=571428.

MAPLE

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

MATHEMATICA

fc[n_]:=Block[{q=RealDigits[1/n][[1, -1]]}, If[IntegerQ[q], 0, While[First[q]==0, q=RotateLeft[q]]; FromDigits[q]]];

Table[fc[n], {n, 36}] (* Ray Chandler, Nov 19 2014, corrected Jun 27 2017 *)

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 *)

CROSSREFS

Cf. A007732, A051626, A002371, A048595, A006883, A007498, A007615, A040017, A051627.