A051626 Period of decimal representation of 1/n, or 0 if 1/n terminates.

0, 0, 1, 0, 0, 1, 6, 0, 1, 0, 2, 1, 6, 6, 1, 0, 16, 1, 18, 0, 6, 2, 22, 1, 0, 6, 3, 6, 28, 1, 15, 0, 2, 16, 6, 1, 3, 18, 6, 0, 5, 6, 21, 2, 1, 22, 46, 1, 42, 0, 16, 6, 13, 3, 2, 6, 18, 28, 58, 1, 60, 15, 6, 0, 6, 2, 33, 16, 22, 6, 35, 1, 8, 3, 1, 18, 6, 6, 13, 0, 9, 5, 41, 6, 16, 21, 28, 2, 44, 1

Offset: 1

Author

J. Lowell

Keywords

• nonn
• base
• easy

Comments

Essentially same as A007732.

For any prime number p: if a(p) > 0, a(p) divides p-1. - David Spitzer, Jan 09 2017

Examples

			From _M. F. Hasler_, Dec 14 2015: (Start)
a(1) = a(2) = 0 because 1/1 = 1 and 1/2 = 0.5 have a finite decimal expansion.
a(3) = a(6) = a(9) = a(12) = 1 because 1/3 = 0.{3}*, 1/6 = 0.1{6}*, 1/9 = 0.{1}*, 1/12 = 0.08{3}* where the sequence of digits {...}* which repeats indefinitely is of length 1.
a(7) = 6 because 1/7 = 0.{142857}* with a period of 6.
a(17) = 16 because 1/17 = 0.{0588235294117647}* with a period of 16.
a(19) = 18 because 1/19 = 0.{052631578947368421}* with a period of 18. (End)
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Project Euler, Reciprocal cycles: Problem 26
• Eric Weisstein's World of Mathematics, Repeating Decimal
• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

Essentially same as A007732. Cf. A002371, A048595, A006883, A036275, A114205, A114206, A001913.

Programs

Maple

A051626 := proc(n) local lpow,mpow ;
    if isA003592(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
                 RETURN(lpow-mpow) ;
              fi ;
          od ;
          lpow := lpow+1 ;
       od ;
    fi ;
end: # R. J. Mathar, Oct 19 2006

Mathematica

r[x_]:=RealDigits[1/x]; w[x_]:=First[r[x]]; f[x_]:=First[w[x]]; l[x_]:=Last[w[x]]; z[x_]:=Last[r[x]];
d[x_] := Which[IntegerQ[l[x]], 0, IntegerQ[f[x]]==False, Length[f[x]], True, Length[l[x]]]; Table[d[i], {i,1,90}] (* Hans Havermann, Oct 19 2006 *)
fd[n_] := Block[{q},q = Last[First[RealDigits[1/n]]];If[IntegerQ[q], q = {}]; Length[q]];Table[fd[n], {n, 100}] (* Ray Chandler, Dec 06 2006 *)
Table[Length[RealDigits[1/n][[1,-1]]],{n,90}] (* Harvey P. Dale, Jul 03 2011 *)
a[n_] := If[ PowerMod[10, n, n] == 0, 0, MultiplicativeOrder[10, n/2^IntegerExponent[n, 2]/5^IntegerExponent[n, 5]]]; Array[a, 90] (* myself in A003592 and T. D. Noe in A007732 *) (* Robert G. Wilson v, Feb 20 2025 *)

PARI

A051626(n)=if(1M. F. Hasler, Dec 14 2015

Python

def A051626(n):
    if isA003592(n):
        return 0
    else:
        lpow=1
        while True:
            for mpow in range(lpow-1,-1,-1):
                if (10**lpow-10**mpow) % n == 0:
                    return lpow-mpow
            lpow += 1 # Kenneth Myers, May 06 2016

Python

from sympy import multiplicity, n_order
def A051626(n): return 0 if (m:=(n>>(~n & n-1).bit_length())//5**multiplicity(5,n)) == 1 else n_order(10,m) # Chai Wah Wu, Aug 11 2022

Formula

a(n)=A132726(n,1); a(n)=a(A132740(n)); a(A132741(n))=a(A003592(n))=0. - Reinhard Zumkeller, Aug 27 2007

Extensions

More terms from James Sellers