A121090 -id:A121090 - cp's OEIS frontend

A007732 Period of decimal representation of 1/n.

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

A291943 a(0)=0; for n>0, a(n) = (2n)-th digit after the decimal point in the decimal expansion of 1/(2n+1).

Original entry on oeis.org

0, 3, 0, 7, 1, 9, 3, 6, 7, 1, 4, 3, 0, 3, 1, 9, 3, 5, 7, 2, 9, 3, 2, 7, 8, 1, 3, 1, 1, 1, 9, 1, 3, 7, 1, 9, 3, 3, 9, 1, 7, 3, 7, 1, 1, 9, 1, 5, 7, 1, 9, 3, 0, 7, 1, 0, 3, 6, 0, 0, 8, 0, 0, 7, 0, 9, 8, 0, 7, 1, 0, 9, 8, 4, 1, 9, 4, 4, 7, 0, 6, 3, 0, 7, 3, 5, 3, 4, 0, 1, 9, 0, 4, 5, 0, 9, 3, 0, 7, 1

Offset: 0

Marco Matosic, Sep 06 2017

			a(3)=7 since we want the sixth decimal digit of 1/7.

John H. Conway & Richard K. Guy, The Book of Numbers; Springer 1996.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A002371, A003060, A006883, A036275, A051626, A061480, A114205, A121090, A270392.

f:= proc(n) floor(10^(2*n)/(2*n+1)) mod 10 end proc:
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Oct 31 2017

f[n_] := Mod[Floor[10^(2n)/(2n +1)], 10]; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Oct 31 2017 *)

Edited by N. J. A. Sloane, Oct 30 2017

a(82) corrected by Robert Israel, Oct 31 2017

A158899 These are numbers n such that the reciprocal, 1/n, is a repeating fraction whose period is n/2 - 1.

Original entry on oeis.org

14, 34, 38, 46, 58, 94, 118, 122, 194, 218, 226, 262, 298, 334, 358, 362, 386, 446, 458, 466, 514, 526, 538, 626, 674, 734, 758, 766, 778, 838, 866, 922, 974, 982, 998, 1006, 1018, 1082, 1142, 1154, 1186, 1238, 1294, 1318, 1402, 1418, 1454, 1486, 1622, 1642

Offset: 1

Robert Hutchins, Mar 29 2009

These numbers relate to the long period primes, those that for 1/m the period is m-1 (sequence A006883) in that by multiplying each term in the long period primes by 2, this sequence is generated.

Cf. A006883, A007732, A085837, A121090.

forstep(n=2, 2e3, 2, if ((setminus(Set(factor(n)[,1]), Set([2,5])) != []) && (znorder(Mod(10, n/2^valuation(n, 2)/5^valuation(n, 5))) + 1 == n/2), print1(n, ", "));); \\ Michel Marcus, Feb 24 2013

More terms and edited by Michel Marcus, Feb 24 2013

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.

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A007732 Period of decimal representation of 1/n.

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A291943 a(0)=0; for n>0, a(n) = (2n)-th digit after the decimal point in the decimal expansion of 1/(2n+1).

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A158899 These are numbers n such that the reciprocal, 1/n, is a repeating fraction whose period is n/2 - 1.

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PARI

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A225488 Murai Chuzen numbers.

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