A128858 -id:A128858 - cp's OEIS frontend

A128857 a(n) = least number m beginning with 1 such that the quotient m/n is obtained merely by shifting the leftmost digit 1 of m to the right end.

1, 105263157894736842, 1034482758620689655172413793, 102564, 102040816326530612244897959183673469387755, 1016949152542372881355932203389830508474576271186440677966

Offset: 1

Anton V. Chupin (chupin(X)icmm.ru), Apr 12 2007

a(n) is simply the decimal period of the fraction n/(10n-1). Thus, we have: n/(10n-1) = a(n)/(10^A128858(n)-1). With the usual convention that the decimal period of 0 is zero, that definition would allow the extension a(0)=0. a(n) is also the period of the decadic integer -n/(10n-1). - Gerard P. Michon, Oct 31 2012

			a(4) = 102564 since this is the smallest number that begins with 1 and which is divided by 4 when the first digit 1 is made the last digit (102564/4 = 25641).

A. V. Chupin, Table of n, a(n) for n=1..101
G. P. Michon, Integers that are divided by k if their digits are rotated left.

Minimal numbers for shifting any digit from the left to the right (not only 1) are in A097717.

By accident, the nine terms of A092697 coincide with the first nine terms of the present sequence. - N. J. A. Sloane, Apr 13 2009

(*Moving digits a:*) Give[a_,n_]:=Block[{d=Ceiling[Log[10,n]],m=(10n-1)/GCD[10n-1, a]}, If[m!=1,While[PowerMod[10,d,m]!=n,d++ ],d=1]; ((10^(d+1)-1) a n)/(10n-1)]; Table[Give[1,n],{n,101}]

from sympy import n_order
def A128857(n): return n*(10**n_order(10,(m:=10*n-1))-1)//m # Chai Wah Wu, Apr 09 2024

Edited by N. J. A. Sloane, Apr 13 2009

Code and b-file corrected by Ray Chandler, Apr 29 2009

A217852 Multiplicative order of 5 (mod 5*n - 1).

Original entry on oeis.org

1, 6, 6, 9, 2, 14, 16, 4, 5, 42, 18, 29, 16, 22, 36, 39, 6, 44, 46, 30, 4, 27, 18, 48, 3, 42, 22, 69, 12, 37, 30, 52, 20, 52, 14, 89, 22, 18, 96, 33, 16, 45, 106, 72, 24, 114, 12, 119, 30, 82, 42, 36, 10, 67, 136, 6, 5, 272, 42, 44, 36, 102, 156, 70, 54, 138, 166

Offset: 1

Arkadiusz Wesolowski, Nov 16 2012

nonn

Least m such that 5*n - 1 divides 5^m - 1.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Order

Cf. A002326, A003572, A003574, A128858.

List([1..70],n->OrderMod(5,5*n-1)); # Muniru A Asiru, Feb 25 2019

Table[MultiplicativeOrder[5, 5*n - 1], {n, 67}]

vector(80, n, znorder(Mod(5, 5*n-1))) \\ Michel Marcus, Feb 09 2015

A366494 a(n) is the number of cycles of the map f(x) = 10x mod (10n - 1).

Original entry on oeis.org

8, 1, 1, 8, 2, 1, 5, 6, 2, 53, 1, 4, 8, 3, 1, 14, 4, 1, 41, 2, 16, 29, 1, 34, 8, 49, 1, 26, 2, 7, 11, 16, 4, 5, 3, 2, 80, 1, 1, 26, 2, 1, 83, 2, 14, 29, 9, 2, 8, 1, 1, 14, 2, 27, 17, 16, 2, 5, 9, 2, 14, 1, 25, 26, 16, 1, 5, 8, 14, 5, 1, 2, 32, 3, 5, 50, 4, 17, 5, 4, 4, 143

Offset: 1

Hillel Wayne, Oct 10 2023

nonn

Taking the length of each orbit that starts from f(0)=1 gives the sequence A128858.
Equivalently, the number of cyclotomic cosets of 10 mod (10*n - 1). See A006694.
Map is the Multiply-with-carry algorithm with a=n, b=10, and c=1.

			For a(4) the 8 cycles are:
  (1 10 22 25 16 4)
  (2 20 5 11 32 8)
  (3 30 27 36 9 12)
  (6 21 15 33 18 24)
  (7 31 37 19 34 28)
  (13)
  (14 23 35 38 29 17)
  (26)

Hillel Wayne, Table of n, a(n) for n = 1..1002
George Marsaglia, Random Number Generators, Journal of Modern Applied Statistical Methods, Volume 2, Issue 1 (2003).
Kenneth Shum, Cyclotomic cosets.

Cf. A006694, A023142, A128858, A128857.

a(n)=sumdiv(10*n-1, d, eulerphi(d)/znorder(Mod(10, d)))-1;
vector(100, n, a(n-1)) \\ Joerg Arndt, Jan 22 2024

def get_num_orbits(n: int) -> int:
    orbits = 0
    mod = 10*n - 1
    seen = set()
    for i in range(1, mod):
        if i not in seen:
            seen.add(i)
            orbits += 1
        x = 10*i % mod
        while x != i:
            seen.add(x)
            x = 10*x % mod
    return orbits

from sympy import totient, n_order, divisors
def A366494(n): return sum(totient(d)//n_order(10,d) for d in divisors(10*n-1,generator=True) if d>1) # Chai Wah Wu, Apr 09 2024

cp's OEIS Frontend

A128857 a(n) = least number m beginning with 1 such that the quotient m/n is obtained merely by shifting the leftmost digit 1 of m to the right end.

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A217852 Multiplicative order of 5 (mod 5*n - 1).

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A366494 a(n) is the number of cycles of the map f(x) = 10x mod (10n - 1).

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cp's OEIS Frontend

A128857 a(n) = least number m beginning with 1 such that the quotient m/n is obtained merely by shifting the leftmost digit 1 of m to the right end.

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A217852 Multiplicative order of 5 (mod 5*n - 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Mathematica

PARI

A366494 a(n) is the number of cycles of the map f(x) = 10*x mod (10*n - 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Python

A366494 a(n) is the number of cycles of the map f(x) = 10x mod (10n - 1).