A051628 -id:A051628 - cp's OEIS frontend

Showing 1-10 of 10 results.

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

Offset: 1

Floor van Lamoen

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

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

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

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

Cf. A007732, A051626, A002371, A048595, A006883, A007498, A007615, A040017, A051627.
See also A060282, A060283, A060251.
A051628 is length of preamble.

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

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 *)

Corrected and extended by N. J. A. Sloane

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

A054710 Number of powers of 10 mod n.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 6, 4, 1, 2, 2, 3, 6, 7, 2, 5, 16, 2, 18, 3, 6, 3, 22, 4, 3, 7, 3, 8, 28, 2, 15, 6, 2, 17, 7, 3, 3, 19, 6, 4, 5, 7, 21, 4, 2, 23, 46, 5, 42, 3, 16, 8, 13, 4, 3, 9, 18, 29, 58, 3, 60, 16, 6, 7, 7, 3, 33, 18, 22, 7, 35, 4, 8, 4, 3, 20, 6, 7, 13, 5, 9, 6, 41, 8, 17, 22, 28, 5, 44, 2

Offset: 1

Henry Bottomley, Apr 20 2000

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A007732, A051628, A121341.

Cf. A054703 (base 2), A054704 (3), A054705 (4), A054706 (5), A054707 (6), A054708 (7), A054709 (8), A054717 (9), A351524 (11), A054712 (12), A054713 (13), A054714 (14), A054715 (15), A054716 (16).

Table[Length[Union[PowerMod[10, Range[0,n], n]]], {n,100}] (* T. D. Noe, Aug 30 2006 *)
a[n_] := Module[{e = IntegerExponent[n, {2, 5}]}, Max[e] + MultiplicativeOrder[10, n/Times @@ ({2, 5}^e)]]; Array[a, 100] (* Amiram Eldar, Aug 25 2024 *)

a(n) = A007732(n) + A051628(n). - Amiram Eldar, Aug 25 2024

A065502 Positive numbers divisible by 2 or 5; 1/n not purely periodic after decimal point.

Original entry on oeis.org

2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 28, 30, 32, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 55, 56, 58, 60, 62, 64, 65, 66, 68, 70, 72, 74, 75, 76, 78, 80, 82, 84, 85, 86, 88, 90, 92, 94, 95, 96, 98, 100, 102, 104, 105, 106, 108, 110, 112, 114

Offset: 1

Len Smiley, Nov 25 2001

Complement of A045572. - Reinhard Zumkeller, Nov 15 2009
Numbers that cannot be prefixed by a single digit to form a prime in decimal representation: A124665 is a subsequence. - Reinhard Zumkeller, Oct 22 2011
Up to 198, this is almost identical to "a(n) = n such that 3^n-1 is not squarefree", with the only exceptions being 39 and 117, which are not in this sequence. Why is that? - Felix Fröhlich, Oct 19 2014
The asymptotic density of this sequence is 3/5. - Amiram Eldar, Mar 09 2021

Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).

Cf. A000035, A001622, A045572, A051628, A079998, A124665, A047229 (numbers divisible by 2 or 3).

a065502 n = a065502_list !! (n-1)
a065502_list = filter ((> 1) . (gcd 10)) [1..]
-- Reinhard Zumkeller, Oct 22 2011

A065502 := proc(n)
     option remember;
     if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if (a mod 2) =0 or (a mod 5) =0 then
                return a;
            end if;
        end do:
    end if;
end proc; # R. J. Mathar, Jul 20 2012

Select[Range[114], Mod[#, 2] == 0 || Mod[#, 5] == 0 &] (* T. D. Noe, Jul 13 2012 *)
Select[ Range@ 114, MemberQ[{0, 2, 4, 5, 6, 8}, Mod[#, 10]] &] (* Robert G. Wilson v, May 22 2014 *)

isok(m) = ! ((m%2) && (m%5)); \\ Michel Marcus, Mar 09 2021

A000035(a(n))*(1-A079998(a(n)))=0. - Reinhard Zumkeller, Nov 15 2009
G.f.: x*(2*x^4+x^2+2) / ((x-1)^2*(x^2-x+1)*(x^2+x+1)). - Colin Barker, Jul 18 2013
a(n) = 10*floor(n/6)+s(n mod 6)-floor(((n-1)mod 6)/5), where s(n) = n+1+floor((n+1)/3). - Gary Detlefs, Oct 05 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/5 + log(phi)/sqrt(5), where phi is the golden ratio (A001622). - Amiram Eldar, Dec 28 2021

Offset changed from 0 to 1 by Harry J. Smith, Oct 20 2009

A121341 Number of decimal places before 1/n either recurs or terminates.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 6, 3, 1, 1, 2, 3, 6, 7, 2, 4, 16, 2, 18, 2, 6, 3, 22, 4, 2, 7, 3, 8, 28, 2, 15, 5, 2, 17, 7, 3, 3, 19, 6, 3, 5, 7, 21, 4, 2, 23, 46, 5, 42, 2, 16, 8, 13, 4, 3, 9, 18, 29, 58, 3, 60, 16, 6, 6, 7, 3, 33, 18, 22, 7, 35, 4, 8, 4, 3, 20, 6, 7, 13, 4, 9, 6, 41, 8, 17, 22, 28, 5, 44, 2, 6

Offset: 1

Anthony C Robin, Aug 29 2006

In this sequence, the repeating decimals (e.g., 1/7) are treated differently from nonrepeating decimals (e.g., 1/5). If they are treated the same, then a(2)=2, a(4)=3, a(5)=2, a(8)=4, a(10)=2, ... and we obtain A054710. The two sequence differ only for n = 2^j * 5^k.

			1/592 = 0.0016891891891... starts with 4 decimals (0016, zeros counted) and has period 3 (digits 891) to yield a(592) = 4 + 3 = 7.

T. D. Noe, Table of n, a(n) for n = 1..1000 (corrected by Sean A. Irvine, Apr 29 2022)
Index entries for sequences related to decimal expansion of 1/n

A007732 is the length of the periods and serves as a lower bound. Cf. A061075.

Cf. A051626, A051628.

a[n_] := Max[IntegerExponent[n, 2], IntegerExponent[n, 5]] + Length[RealDigits[1/n][[1, -1]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jul 20 2022 *)

a(n) = A051628(n) + A051626(n). - Sean A. Irvine, Apr 13 2022

More terms from T. D. Noe, Aug 30 2006

Additional comments from R. J. Mathar, Aug 30 2006

A114206 Length of the preperiodic part of the decimal expansion of 1/n, and also leading zeros from the period part if the preperiodic digits are all 0s (if any).

Original entry on oeis.org

1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2

Offset: 2

N. J. A. Sloane, Oct 17 2006

Differs from A386406 at n = 92, 208, 248...

			For n = 11, 1/n = 0.(09), so the preperiodic part is empty and the initial 0 of the periodic part is included for length a(11) = 1.
For n = 92, 1/n = 0.01(0869565217391304347826) , so the preperiodic part is "01" and has length a(92) = 2 and the initial 0 in the periodic part is not included since that preperiodic part is not all 0s (unlike the way A386406(92) = 3 does include that inat periodic initial 0).

Hans Havermann, Table of n, a(n) for n = 2..10000
Index entries for sequences related to decimal expansion of 1/n.

Cf. A004216, A051628, A114205, A386406.

fb[n_] := Block[{rd, o, p},rd = RealDigits[1/n];o = Last[rd];p = First[rd];If[ ! IntegerQ[Last[p]], p = Most[p]];Length[p] - o];Table[fb[n], {n, 120}] (* Ray Chandler, Oct 18 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]]; b[x_]:=Which[IntegerQ[l[x]], Length[w[x]]-1*z[x], IntegerQ[f[x]]==False, -1*z[x], True, Length[Drop[w[x],-1]]-1*z[x]]; (* Hans Havermann, Oct 18 2006 *)
Table[b[i], {i,2,128}]

a(n) = max(logint(n,10), max(valuation(n,2), valuation(n,5))); \\ Kevin Ryde, Jul 22 2025

a(n) = max(A004216(n), A051628(n)). - Kevin Ryde, Jul 22 2025

More terms from Ray Chandler and Hans Havermann, Oct 18 2006

Edited by Andrei Zabolotskii and Kevin Ryde, Jul 20 2025

A386406 Length of the preperiodic part of the decimal expansion of 1/n, including any leading zeros from the period.

Original entry on oeis.org

Offset: 2

Andrei Zabolotskii, Jul 20 2025

See A114205.

			For n = 92, 1/n = 0.01(0869565217391304347826) = 0.010(8695652173913043478260), so the preperiodic part is "010" and has length a(92) = 3.

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

Cf. A051628, A114205, A114206.

b[n_] := Block[{p,o}, {p,o} = RealDigits[1/n]; If[!IntegerQ[Last[p]], p = Join[Most[p],TakeWhile[Last[p],#==0&]]]; Length[p]-o];
Table[b[n], {n,2,100}]

a(n) = my(pre = max(valuation(n,2),valuation(n,5)), r = 10^pre % n); pre + if(r,logint(n\r,10)); \\ Kevin Ryde, Jul 22 2025

a(n) = p + (floor(log_10(1/f)) if f!=0), where p = A051628(n) and f = frac(10^p/n). - Kevin Ryde, Jul 22 2025

A276981 Irregular triangle T(n,k) read by rows of residue classes of powers of 10 modulo n.

Original entry on oeis.org

0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 4, 1, 3, 2, 6, 4, 5, 1, 2, 4, 0, 1, 1, 0, 1, 10, 1, 10, 4, 1, 10, 9, 12, 3, 4, 1, 10, 2, 6, 4, 12, 8, 1, 10, 1, 10, 4, 8, 0, 1, 10, 15, 14, 4, 6, 9, 5, 16, 7, 2, 3, 13, 11, 8, 12, 1, 10, 1, 10, 5, 12, 6, 3, 11, 15, 17, 18, 9, 14, 7, 13, 16, 8, 4, 2

Offset: 1

Martin Renner, Apr 11 2017

The length of the nonperiodic part of the residue class values is given in A051628, the length of the periodic part is given in A007732.

These residue class values are useful to check the divisibility of a number by the divisor n simply by calculating the weighted sum of digits. For example, the number 86415 is divisible by 7, because the weighted sum of digits 5*1 + 1*3 + 4*2 + 6*6 + 8*4 = 84 is divisible by 7. The used weights are the residue class values for n = 7: 1, 3, 2, 6, 4, 5, ... for ones, tens, hundreds, ...

			T(n,k), 1 <= k <= A051628(n) + A007732(n), starts with
n = 1:  0
n = 2:  1, 0
n = 3:  1
n = 4:  1, 2, 0
n = 5:  1, 0
n = 6:  1, 4
n = 7:  1, 3, 2, 6, 4, 5
n = 8:  1, 2, 4, 0
n = 9:  1
n = 10: 1, 0
n = 11: 1, 10
n = 12: 1, 10, 4
etc.

Alois P. Heinz, Rows n = 1..800, flattened

Cf. A007732, A051628.

a:=proc(n)
  local R,N,P,i;
  R:=[seq(10^k mod n,k=0..n)]; # residue class
  N:=[]; # nonperiodic part
  P:=[]; # periodic part
  for i from 1 to nops(R) do
    member(R[i],R,'m');
    if m

A307070 a(n) is the number of decimal places before the decimal expansion of 1/n terminates, or the period of the recurring portion of 1/n if it is recurring.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 6, 3, 1, 1, 2, 1, 6, 6, 1, 4, 16, 1, 18, 2, 6, 2, 22, 1, 2, 6, 3, 6, 28, 1, 15, 5, 2, 16, 6, 1, 3, 18, 6, 3, 5, 6, 21, 2, 1, 22, 46, 1, 42, 2, 16, 6, 13, 3, 2, 6, 18, 28, 58, 1, 60, 15, 6, 6, 6, 2, 33, 16, 22, 6, 35, 1, 8, 3, 1, 18, 6, 6, 13

Offset: 1

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Author

Luke W. Richards, Mar 22 2019

Keywords

nonn
base
easy

Comments

If the decimal expansion of 1/n terminates, we will write it as ending with infinitely many 0's (rather than 9's). Then for any n > 1, the expansion of 1/n consists of a preamble whose length is given by A051628(n), followed by a periodic part with period length A007732(n). This sequence is defined as follows: If the only primes dividing n are 2 and 5 (see A003592), a(n) = A051628(n), otherwise a(n) = A007732(n) (and the preamble is ignored). - N. J. A. Sloane, Mar 22 2019

This sequence was discovered by a school class (aged 12-13) at Arden School, Solihull, UK.

Equally space the digits 0-9 on a circle. The digits of the decimal expansion of rational numbers can be connected on this circle to form data visualizations. This sequence is useful, cf. A007732 or A051626, for identifying the complexity of that visualization.

Examples

1/1 is 1.0. There are no decimal digits, so a(1) = 0. 1/2 is 0.5. This is a terminating decimal. There is 1 digit, so a(2) = 1. 1/6 is 0.166666... This is a recurring decimal with a period of 1 (the initial '1' does not recur) so a(6) = 1. 1/7 is 0.142857142857... This is a recurring decimal, with a period of 6 ('142857') so a(7) = 6.

Crossrefs

See A114205 and A051628 for the preamble, A036275 and A051626 for the periodic part.

Cf. A001913, A003592, A054710, A121341, A122060.

Programs

PARI

a(n) = my (t=valuation(n,2), f=valuation(n,5), r=n/(2^t*5^f)); if (r==1, max(t,f), znorder(Mod(10, r))) \\ Rémy Sigrist, May 08 2019

Python

def sequence(n): count = 0 dividend = 1 remainder = dividend % n remainders = [remainder] no_recurrence = True while remainder != 0: count += 1 dividend = remainder * 10 remainder = dividend % n if remainder in remainders: if no_recurrence: no_recurrence = False remainders = [remainder] else: return len(remainders) else: remainders.append(remainder) else: return count

Extensions

More terms from Rémy Sigrist, May 08 2019

A357756 a(n) is the least k > 0 such that A007953(n*k) equals A007953((n*k)^2), where A007953 is the sum of the digits.

Original entry on oeis.org

1, 1, 5, 3, 25, 2, 3, 27, 62, 1, 1, 5, 15, 27, 128, 3, 31, 17, 1, 1, 5, 9, 9, 2, 75, 4, 18, 7, 64, 5, 3, 16, 56, 3, 85, 17, 5, 27, 5, 9, 25, 9, 45, 13, 27, 1, 1, 27, 66, 54, 2, 9, 9, 18, 22, 1, 32, 15, 25, 135, 3, 18, 8, 3, 28, 9, 3, 43, 47, 72, 27, 8, 25, 126, 27

Offset: 0

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Author

Thomas Scheuerle, Oct 12 2022

Keywords

nonn
base

Comments

A task in the German competition "Bundeswettbewerb Mathematik 2021" was to prove that for each positive integer n there exists a k such that A007953(n*k) = A007953((n*k)^2).

One of the proposed proofs uses the argument that numbers of the form m = (10^x-1)*(10^y) will have the desired property A007953(m) = A007953(m^2). Thus we need to prove that we can find for all n a k, x and y such that n*k = (10^x-1)*(10^y). Let n be of the form b*2^c*5^d with b odd and not divisible by 5, then we know that y = max(c, d). From Euler's totient theorem we know that 10^x-1 will be divisible by e if x = A000010(e) where A000010 is Euler's totient function. See the formula section for the corresponding resulting k.

a(n) will never be divisible by 10.

If n is divisible by 3 but not by 9, then a(n) is divisible by 3. - Robert Israel, Oct 13 2022

Links

Johann Beurich, Wie kann man das beweisen? (Bundeswettbewerb Mathematik 2021), YouTube video, 2021 (in German).

Bundeswettbewerb Mathematik 2021, Die Aufgaben_und_Loesungen der 2. Runde 2021 (in German). The proof and solutions.

Crossrefs

Cf. A000010, A007953, A051628, A058369, A060284, A132740, A178505.

Programs

Maple

f:= proc(n) local k; for k from 1 do if sd(n*k) = sd((n*k)^2) then return k fi od end proc: map(f, [$1..100]); # Robert Israel, Oct 13 2022

PARI

a(n) = {my(k = 1); while(sumdigits(n*k)!=sumdigits((n*k)^2),k++);k}

Python

def sd(n): return sum(map(int, str(n))) def a(n): k = 1 while not sd(n*k) == sd((n*k)**2): k += 1 return k print([a(n) for n in range(75)]) # Michael S. Branicky, Oct 13 2022

Formula

a(A058369(n)) = 1.

a(a(n)) <= n.

a(n) <= A132740(n)*A060284(A132740(n))*10^A051628(n)/n.

or a(n) <= (10^A000010(A132740(n))-1)*10^A051628(n)/n.

A384869 For n >= 1, a(n) = Sum_{k = 1..n} gcd(n, floor((n/k)*10^x)), where x = A121341(k/gcd(n,k)).

Original entry on oeis.org

1, 3, 7, 8, 17, 21, 31, 27, 53, 33, 71, 58, 85, 74, 103, 75, 129, 118, 145, 70, 209, 141, 199, 146, 197, 194, 309, 191, 281, 175, 301, 206, 427, 271, 339, 297, 397, 306, 503, 157, 481, 432, 505, 336, 559, 395, 553, 388, 607, 303, 777, 454, 677, 620, 605, 467

Offset: 1

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Author

Ctibor O. Zizka, Jun 11 2025

Keywords

base
nonn

Comments

a(n) < n^2 - n + 1.

Examples

For n = 12: k = 4, x = A121341(4/gcd(12,4)) = 0, gcd(12, floor((12/4)*10^0)) = 3; k = 5, x = A121341(5/gcd(12,5)) = 1, gcd(12, floor((12/5)*10^1)) = 12; and so on.

Crossrefs

Cf. A003592, A018804, A051626, A051628, A085837, A121341.

Programs

Mathematica

f[n_] := Max[IntegerExponent[n, 2], IntegerExponent[n, 5]] + Length[RealDigits[1/n][[1, -1]]]; a[n_] := Sum[GCD[n, Floor[(n/k)*10^f[k/GCD[n, k]]]], {k, 1, n}]; Array[a, 100] (* Amiram Eldar, Jun 19 2025 *)

Showing 1-10 of 10 results.

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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.

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Maple

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A054710 Number of powers of 10 mod n.

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Mathematica

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A065502 Positive numbers divisible by 2 or 5; 1/n not purely periodic after decimal point.

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Haskell

Maple

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A121341 Number of decimal places before 1/n either recurs or terminates.

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Mathematica

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A114206 Length of the preperiodic part of the decimal expansion of 1/n, and also leading zeros from the period part if the preperiodic digits are all 0s (if any).

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Mathematica

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A386406 Length of the preperiodic part of the decimal expansion of 1/n, including any leading zeros from the period.

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Mathematica

PARI

Formula

A276981 Irregular triangle T(n,k) read by rows of residue classes of powers of 10 modulo n.

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A357756 a(n) is the least k > 0 such that A007953(nk) equals A007953((nk)^2), where A007953 is the sum of the digits.