A036275 -id:A036275 - cp's OEIS frontend

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

J. Lowell

Essentially same as A007732.

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

			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)

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

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

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

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 *)

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

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

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

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

More terms from James Sellers

A060284 Periodic part of decimal expansion of 1/n (leading 0's omitted).

Original entry on oeis.org

0, 0, 3, 0, 0, 6, 142857, 0, 1, 0, 9, 3, 76923, 714285, 6, 0, 588235294117647, 5, 52631578947368421, 0, 47619, 45, 434782608695652173913, 6, 0, 384615, 37, 571428, 344827586206896551724137931, 3, 32258064516129, 0, 3, 2941176470588235, 285714, 7, 27, 263157894736842105

Offset: 1

N. J. A. Sloane, Mar 30 2001

			1/11 = .09090909..., so a(11) = 9.

Ray Chandler, Table of n, a(n) for n = 1..1000 (first 500 terms from T. D. Noe)
Index entries for sequences related to decimal expansion of 1/n

Cf. A036275, A060283, A060251.

More terms from Klaus Brockhaus and Jason Earls, Mar 30 2001
Offset corrected by R. J. Mathar, Jun 26 2010
B-file extended and a(168) and a(184) corrected by Ray Chandler, Jun 27 2017

A060283 Periodic part of decimal expansion of reciprocal of n-th prime (leading 0's moved to end).

Original entry on oeis.org

0, 3, 0, 142857, 90, 769230, 5882352941176470, 526315789473684210, 4347826086956521739130, 3448275862068965517241379310, 322580645161290, 270, 24390, 232558139534883720930, 2127659574468085106382978723404255319148936170

Offset: 1

N. J. A. Sloane, Mar 30 2001

			1/11 = .09090909..., so a(5) = 90.

T. D. Noe, Table of n, a(n) for n=1..100

Cf. A036275, A060251.

Table[FromDigits[FindTransientRepeat[RealDigits[1/p,10,100][[1]],2][[2]]],{p,Prime[Range[20]]}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 10 2021 *)

a(n)=t=iferr(znorder(Mod(10,n)),E,0);d=(10^t-1)/n;s=t-#Str(d);if(s,d*10^s,d)
forprime(i=1,1e2,print1(a(i)", ")) \\ Lear Young, Mar 01 2014

A060283 = A036275 o A000040, i.e., a(n) = A036275(A000040(n)). - M. F. Hasler, Dec 28 2015

More terms from Klaus Brockhaus, Mar 30 2001

A114205 Write decimal expansion of 1/n as 0.PPP...PQQQ..., where QQQ... is the cyclic part. If the expansion does not terminate, any leading 0's in QQQ... are regarded as being at the end of the PPP...P part. Sequence gives PPP...P, right justified, with leading zeros omitted.

Original entry on oeis.org

5, 0, 25, 2, 1, 0, 125, 0, 1, 0, 8, 0, 0, 0, 625, 0, 0, 0, 5, 0, 0, 0, 41, 4, 0, 0, 3, 0, 0, 0, 3125, 0, 0, 0, 2, 0, 0, 0, 25, 0, 0, 0, 2, 0, 0, 0, 208, 0, 2, 0, 1, 0, 0, 0, 17, 0, 0, 0, 1, 0, 0, 0, 15625, 0, 0, 0, 1, 0, 0, 0, 13, 0, 0, 1, 1, 0, 0, 0, 125, 0, 0, 0, 1, 0, 0, 0, 11, 0, 0, 0, 10

Offset: 2

N. J. A. Sloane, Oct 17 2006

b(n) = A386406(n) gives the length of P (including leading zeros), c(n) = A036275(n) gives the smallest cycle in QQQ... (including terminating zeros) and d(n) = A051626(n) gives the length of that cycle.

Thus 1/n = 10^(-b(n)) * ( a(n) + c(n)/(10^d(n) - 1) ). When c(n)=d(n)=0, the fraction c(n)/(10^d(n) - 1), which is 0/0, evaluates (by definition) to 0.

			n .. expansion of 1/n .... a b c d
2 .50000000000000000000... 5 1 0 0
3 .33333333333333333333... 0 0 3 1
4 .25000000000000000000... 25 2 0 0
5 .20000000000000000000... 2 1 0 0
6 .16666666666666666667... 1 1 6 1
7 .14285714285714285714... 0 0 142857 6
8 .12500000000000000000... 125 3 0 0
9 .11111111111111111111... 0 0 1 1
10 .1000000000000000000... 1 1 0 0
11 .0909090909090909090... 0 1 90 2
12 .0833333333333333333... 8 2 3 1
13 .0769230769230769230... 0 1 769230 6
14 .0714285714285714285... 0 1 714285 6
15 .0666666666666666666... 0 1 6 1
16 .0625000000000000000... 625 4 0 0
(Start)
92 .0108695652173913043... 10  3 869...260 22
102 .009803921568627450... 0   2 980...450 16
416 .002403846153846153... 240 5 384615    6
4544 .00022007042253521... 2200 7 704...450 35
(End) - _Ruud H.G. van Tol_, Nov 20 2024

Cf. A386406, A114206, A036275, A051626, A060284, A007732, A175555.

A114205 := proc(n) local sh,lpow,mpow,a,b ; lpow:=1 ; while true do for mpow from lpow-1 to 0 by -1 do if (10^lpow-10^mpow) mod n =0 then a := (10^lpow-10^mpow)/n ; sh := 10^(lpow-mpow)-1 ; b := a mod sh ; a := floor(a/sh) ; while b>0 and b*10 < sh+1 do a := 10*a ; b := 10*b ; end ; RETURN(a) ; fi ; od ; lpow := lpow+1 ; od ; end: for n from 2 to 600 do printf("%d %d ",n,A114205(n)) ; od ; # R. J. Mathar, Oct 19 2006

fa[n_] := Block[{p},p = First[RealDigits[1/n]];If[ ! IntegerQ[Last[p]], p =  Join[Most[p],TakeWhile[Last[p],#==0&]]];FromDigits[p]];Table[fa[n], {n, 100}] (* Ray Chandler, Oct 18 2006 *)

a(n)= my(s=max(valuation(n, 2), valuation(n, 5))); s||return(0); my([p, r]= divrem(10^s, n)); if(r&&(r=n\r)>9, s+=logint(r, 10)); 10^s\n; \\ Ruud H.G. van Tol, Nov 19 2024

More terms from Ray Chandler and Hans Havermann, Oct 18 2006
I would also like to get programs that produce this and A114206, A036275, A051626 in Maple.
Edited by Andrei Zabolotskii, Jul 20 2025

A178505 Decimal form of the period of 1/n for n such that gcd(n,10)=1. Leading zeros are suppressed.

Original entry on oeis.org

3, 142857, 1, 9, 76923, 588235294117647, 52631578947368421, 47619, 434782608695652173913, 37, 344827586206896551724137931, 32258064516129, 3, 27, 25641, 2439, 23255813953488372093

Offset: 1

Michel Lagneau, May 29 2010

The numbers n are A045572, and the corresponding periods are A002329.

			3 is in the sequence because 1/3 = 0.3333...
142857 is in the sequence because 1/7 = 0.142857 142857 ...
1 is in the sequence because 1/9 = 0.1111....

Ray Chandler, Table of n, a(n) for n = 1..406 (terms up to 1000 digits)

Cf. A002329, A036275, A045572, A060284.

with(numtheory): nn:= 100: T:=array(1..nn):k:=1: U:=array(1..nn):k:=1: for n from 2 to 200 do:x:=1/n:for p from 1 to 200 while(irem(10^p,n)<>1 or gcd(n,10)<> 1) do:od: if irem(10^p,n) = 1 and gcd(n,10) = 1 then y:=floor(x*10^p): T[k]:=y: U[k]:=n : k:=k+1:else fi:od:print(T):

a(n) = A060284(A045572(n+1)). [R. J. Mathar, Jun 26 2010]

Name corrected by T. D. Noe, Jul 07 2010

A060251 a(n) = periodic part of decimal expansion of n/n-th prime (leading 0's moved to end).

Original entry on oeis.org

0, 6, 0, 571428, 45, 461538, 4117647058823529, 421052631578947368, 3913043478260869565217, 3448275862068965517241379310, 354838709677419, 324, 31707, 325581395348837209302, 3191489361702127659574468085106382978723404255

Offset: 1

Jason Earls, Mar 21 2001

			a(4) = 4/7=.571428571428... so a(4)= 571428.

Cf. A060282, A060283, A036275.

More terms from Klaus Brockhaus, Mar 31 2001

Description clarified by Ray Chandler, Jun 27 2017

A175555 Preperiodic part of the decimal expansion of 1/k as k runs through A065502.

Original entry on oeis.org

5, 25, 2, 1, 125, 1, 8, 0, 0, 625, 0, 5, 0, 41, 4, 0, 3, 0, 3125, 0, 0, 2, 0, 25, 0, 2, 0, 0, 208, 2, 1, 0, 0, 17, 0, 1, 0, 15625, 0, 0, 1, 0, 13, 0, 1, 1, 0, 125, 0, 1, 0, 0, 11, 0, 1, 0, 0, 1041, 0, 1

Offset: 1

Michel Lagneau, Jun 29 2010

Multiples of 2 or 5 generate a quotient with a preperiodic sequence of digits, for example 1/24 = 0.041666666..., and 41 is the decimal form of the preperiodic part.

Usually a(n) = A114205(A065502(n)), but the convention in A114205 that leading zeros in the periodic part are attached to the preperiodic part seems not to be used here. - R. J. Mathar, Jul 20 2012

			a(14)=4 is in the sequence because 1/25 = 0.040000... and 4 is the prefix.
208 is in the sequence because 1/48 = 2083333.... and 208 is the prefix.

Cf. A036275.

A175555 := proc(n)
        local k,s,al ;
        k := A065502(n) ;
        for s from 1 do
                for al from 0 to s-1 do
                        if (10^s-10^al) mod k = 0 then
                                return floor(10^al/k) ;
                        end if;
                end do:
        end do:
end proc: # R. J. Mathar, Jul 22 2012

A060282 Periodic part of decimal expansion of reciprocal of n-th prime (leading 0's omitted).

Original entry on oeis.org

0, 3, 0, 142857, 9, 76923, 588235294117647, 52631578947368421, 434782608695652173913, 344827586206896551724137931, 32258064516129, 27, 2439, 23255813953488372093, 212765957446808510638297872340425531914893617

Offset: 1

N. J. A. Sloane, Mar 30 2001

			1/7 = 0.142857142..., so a(4) = 142857.
1/11 = 0.09090909..., so a(5) = 9.

T. D. Noe, Table of n, a(n) for n=1..100

Cf. A002371, A036275, A060283, A060251, A036275, A001913.

primePer[1] = primePer[3] = 0; primePer[n_] := FromDigits[(d = RealDigits[1/Prime[n]])[[1, 1]]] * 10^d[[2]]; Array[ primePer, 15] (* Amiram Eldar, Apr 28 2020 *)

f(n)=if(n<4,n==2,znorder(Mod(10, prime(n)))) \\ A002371
for(n=1,100,print1(floor(10^f(n)/prime(n)),","))

a(n) = floor(10^A002371(n)/prime(n)).
a(n) = 0 if and only if n = 1 or 3, corresponding to the primes 2 and 5, which are factors of 10. - Alonso del Arte, Apr 03 2020
ceiling(log_10(a(n))) = prime(n) - 1 if prime(n) is a full reptend prime (A001913). - Alonso del Arte, Apr 14 2020

More terms from Klaus Brockhaus, Mar 30 2001

A104013 First digit-cycle of binary expansion of 1/n. Any initial 0's are to be placed at end of cycle.

Original entry on oeis.org

0, 0, 10, 0, 1100, 10, 100, 0, 111000, 1100, 1011101000, 10, 100111011000, 100, 1000, 0, 11110000, 111000, 110101111001010000, 1100, 110000, 1011101000, 10110010000, 10, 10100011110101110000, 100111011000, 100101111011010000

Offset: 1

Bryan Jacobs (bryanjj(AT)gmail.com), Feb 25 2005

			1/5 = 0.00110011001100... in binary, so a(5) = 1100.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..1018

Cf. A007733, A036275, A104014.

f[n_] := If[IntegerQ@ Log2@ n, 0, FromDigits[ RealDigits[1/n, 2][[1, 1]]]]; Array[f, 27] (* Robert G. Wilson v, Sep 01 2015 *)

A175562 The periodic part of the decimal expansion of 1/Fibonacci(n) with any initial zeros placed at the end of the cycle.

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 769230, 476190, 2941176470588235, 18, 11235955056179775280898876404494382022471910, 4

Offset: 1

Michel Lagneau, Jul 02 2010

A curiosity: the first six digits (with the first digit zero) of a(11): {0,1,1,2,3,5} are the first six Fibonacci numbers!

The next term of this sequence contains 232 digits (decimal form of the periodic part of 1/233 = 0.0042918454935622317596566523605150214...7210300).

			1/Fibonacci(7) = 1/13 = 0.0769230769230769230... and digit-cycle is 769230, so a(7)= 769230.

Vincenzo Librandi, Table of n, a(n) for n = 1..22

Cf. A036275.

fc[n_] := Block[{q}, q = Last[First[RealDigits[1/Fibonacci[n]]]]; If[IntegerQ[q], q = {}]; FromDigits[q]]; Table[fc[n], {n, 40}] (* see the Mathematica program in A036275 *)

Name and comment corrected by T. D. Noe, Jul 06 2010

cp's OEIS Frontend

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

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A060284 Periodic part of decimal expansion of 1/n (leading 0's omitted).

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A060283 Periodic part of decimal expansion of reciprocal of n-th prime (leading 0's moved to end).

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A114205 Write decimal expansion of 1/n as 0.PPP...PQQQ..., where QQQ... is the cyclic part. If the expansion does not terminate, any leading 0's in QQQ... are regarded as being at the end of the PPP...P part. Sequence gives PPP...P, right justified, with leading zeros omitted.

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A178505 Decimal form of the period of 1/n for n such that gcd(n,10)=1. Leading zeros are suppressed.

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A060251 a(n) = periodic part of decimal expansion of n/n-th prime (leading 0's moved to end).

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A175555 Preperiodic part of the decimal expansion of 1/k as k runs through A065502.

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A060282 Periodic part of decimal expansion of reciprocal of n-th prime (leading 0's omitted).

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