A121341 -id:A121341 - cp's OEIS frontend

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

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

Ctibor O. Zizka, Jun 11 2025

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

			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.

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

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 *)

A007732 Period of decimal representation of 1/n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Hal Sampson [ hals(AT)easynet.com ]

Appears to be a divisor of A007733*A007736. - Henry Bottomley, Dec 20 2001
Primes p such that a(p) = p-1 are in A001913. - Dmitry Kamenetsky, Nov 13 2008
When 1/n has a finite decimal expansion (namely, when n = 2^a*5^b), a(n) = 1 while A051626(n) = 0. - M. F. Hasler, Dec 14 2015
a(n.n) >= a(n) where n.n is A020338(n). - Davide Rotondo, Jun 13 2024

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, pp. 159 etc.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Project Euler, Reciprocal cycles: Problem 26
Index entries for sequences related to decimal expansion of 1/n

Cf. A121341, A066799, A121090, A001913, A084680.

A007732 := proc(n)
    a132740 := 1 ;
    for pe in ifactors(n)[2] do
        if not op(1,pe) in {2,5} then
            a132740 := a132740*op(1,pe)^op(2,pe) ;
        end if;
    end do:
    if a132740 = 1 then
        1 ;
    else
        numtheory[order](10,a132740) ;
    end if;
end proc:
seq(A007732(n),n=1..50) ; # R. J. Mathar, May 05 2023

Table[r = n/2^IntegerExponent[n, 2]/5^IntegerExponent[n, 5]; MultiplicativeOrder[10, r], {n, 100}] (* T. D. Noe, Oct 17 2012 *)

a(n)=znorder(Mod(10,n/2^valuation(n,2)/5^valuation(n,5))) \\ Charles R Greathouse IV, Jan 14 2013

from sympy import n_order, multiplicity
def A007732(n): return n_order(10,n//2**multiplicity(2,n)//5**multiplicity(5,n)) # Chai Wah Wu, Feb 07 2022

def a(n):
    n = ZZ(n)
    rad = 2**n.valuation(2) * 5**n.valuation(5)
    return Zmod(n // rad)(10).multiplicative_order()
[a(n) for n in range(1, 20)]
# F. Chapoton, May 03 2020

Note that if n=r*s where r is a power of 2 and s is odd then a(n)=a(s). Also if n=r*s where r is a power of 5 and s is not divisible by 5 then a(n) = a(s). So we just need a(n) for n not divisible by 2 or 5. This is the smallest number m such that n divides 10^m - 1; m is a divisor of phi(n), where phi = A000010.
phi(n) = n-1 only if n is prime and since a(n) divides phi(n), a(n) can only equal n-1 if n is prime. - Scott Hemphill (hemphill(AT)alumni.caltech.edu), Nov 23 2006
a(n)=a(A132740(n)); a(A132741(n))=a(A003592(n))=1. - Reinhard Zumkeller, Aug 27 2007

More terms from James Sellers, Feb 05 2000

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

A122060 Position in decimal expansion of 1/n where repetition begins.

Original entry on oeis.org

2, 3, 2, 4, 3, 3, 7, 5, 2, 3, 3, 4, 7, 8, 3, 6, 17, 3, 19, 4, 7, 4, 23, 5, 4, 8, 4, 11, 29, 3

Offset: 1

Ben Paul Thurston, Sep 14 2006

If 1/n = 0.XYYYYY... then sequence gives index of first digit of the second Y.

a(4) = 4 and a(p) = p for primes p = {7, 17, 19, 23, 29, 47, 59, 61, 97, ...} = A001913(n) Cyclic numbers: primes with primitive root 10. - Alexander Adamchuk, Jan 28 2007

			a(4) = 4 because in 0.2500 the zero begins repeating at the fourth position.
a(17) = 17 because 0.05882352941176470588... begins repeating at the 17th position.

Cf. A001913, A002371.

a(n)=A121341(n)+2 if 1/n terminates, else a(n)=A121341(n)+1. - R. J. Mathar, Sep 20 2006

A225488 Murai Chuzen numbers.

Original entry on oeis.org

9, 45, 3, 225, 18, 15, -1, 1125, 1, 99, 495, 33, 2475, 198, 165, -1, 12375, 11, 999, 4995, 333, 24975, 1998, 1665, -1, 124875, 111, 9999, 49995, 3333, 249975, 19998, 16665, -1, 1249875, 1111, 99999, 49995, 33333, 2499975, 199998, 166665, -1, 12499875, 11111, 999999, 4999995, 333333, 24999975, 1999998, 1666665, -1, 124999875, 111111

Offset: 1

Jonathan Sondow, May 10 2013

"Murai Chuzen divides 9 by 1, 2, 3, 4, 5, 6, 7, 8, 9, getting the figures 9, 45, 3, 225, 18, 15, x (not divisible), 1125, 1, -- without reference to the decimal points. Similarly he divides 99 by 1, 2, 3, 4, 5, 6, 7, 8, 9, getting the figures 99, 495, 33, 2475, 198, 165, x, 12375, 11. Next he divides 999 by 1, 2, 3, 4, 5, 6, 7, 8, 9, getting the figures 999, 4995, 333, 24975, 1998, 1665, x, 124875, 111." (Smith and Mikami, expanded and corrected)
Smith and Mikami put "x" whenever a decimal does not terminate. In the data, I put -1 instead of "x".
Murai Chuzen concludes that if 1 is divided by 9, 45, 3, 225, 18, 15, 1125, and 1, the results will have one-digit repetends; if 1 is divided by 99, 495, 33, 2475, 198, 165, 12375, and 11, the results will have two-digit repetends; if 1 is divided by 999, 4995, 333, 24975, 1998, 1665, 124875, and 111, the results will have three-digit repetends; etc.

			9/1 = 9, so a(1) = 9; 9/2 = 4.5, so a(2) = 45; 9/7 does not terminate, so a(7) = -1; 9/8 = 1.125, so a(8) = 1125; 9/9 = 1, so a(9) = 1.
99/1 = 99, so a(10) = 99; 99/2 = 49.5, so a(11) = 495.

Murai Chuzen, Sampo Doshi-mon (Arithmetic for the Young), 1781.

David Eugene Smith and Yoshio Mikami, A history of Japanese mathematics, Open Court, 1914, reprinted by Dover, 2004, p. 176.

Cf. A001913, A007732, A066799, A096688, A121090, A121341, A181431.

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

Luke W. Richards, Mar 22 2019

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.

			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.

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

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

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

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

More terms from Rémy Sigrist, May 08 2019

cp's OEIS Frontend

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

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A007732 Period of decimal representation of 1/n.

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A054710 Number of powers of 10 mod n.

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A122060 Position in decimal expansion of 1/n where repetition begins.

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A225488 Murai Chuzen numbers.

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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.

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