A128857 -id:A128857 - cp's OEIS frontend

A128858 Number of digits in A128857(n).

1, 18, 28, 6, 42, 58, 22, 13, 44, 2, 108, 48, 21, 46, 148, 13, 78, 178, 6, 99, 18, 8, 228, 7, 41, 6, 268, 15, 272, 66, 34, 28, 138, 112, 116, 179, 5, 378, 388, 18, 204, 418, 6, 219, 32, 48, 66, 239, 81, 498, 508, 43, 506, 42, 60, 42, 284, 192, 90, 299, 84, 618, 48, 35

Offset: 1

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

A. V. Chupin, Table of n, a(n) for n = 1..1000

Cf. A128857.

Give[a_,n_]:=Block[{d=Floor[Log[10,n]]+1,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)]; Length[IntegerDigits[Give[1,n]]]

from sympy import n_order
def A128858(n): return n_order(10,10*n-1) # Chai Wah Wu, Apr 09 2024

a(n) = order of 10 (mod 10*n - 1). - Arkadiusz Wesolowski, Nov 17 2012

a(10) = 2 corrected by Gerard P. Michon, Oct 31 2012

A097717 a(n) = least number m such that the quotient m/n is obtained merely by shifting the leftmost digit of m to the right end.

Original entry on oeis.org

1, 105263157894736842, 1034482758620689655172413793, 102564, 714285, 1016949152542372881355932203389830508474576271186440677966, 1014492753623188405797, 1012658227848, 10112359550561797752808988764044943820224719

Offset: 1

Lekraj Beedassy, Sep 21 2004

			We have a(5)=714285 since 714285/5=142857.
Likewise, a(4)=102564 since this is the smallest number followed by 205128, 307692, 410256, 512820, 615384, 717948, 820512, 923076, ... which all get divided by 4 when the first digit is made last.

R. Sprague, Recreation in Mathematics, Problem 21 pp. 17; 47-8 Dover NY 1963.

A. V. Chupin, Table of n, a(n) for n=1..101
A. V. Chupin, Table of n, a(n) for n=1..154

A097717: when move L digit to R, divides by n (infinite)
A094676: when move L digit to R, divides by n, no. of digits is unchanged (finite)
A092697: when move R digit to L, multiplies by n (finite)
A128857 is the same sequence as A097717 except that m must begin with 1.
Not the same as A092697.
Cf. A249596 - A249599 (bases 2 to 5).

Min[Table[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)], {a,9}]] (* Anton V. Chupin (chupin(X)icmm.ru), Apr 12 2007 *)

a(9) from Anton V. Chupin (chupin(X)icmm.ru), Apr 12 2007

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

A092697 For 1 <= n <= 9, a(n) = least number m such that the product n*m is obtained merely by shifting the rightmost digit of m to the left end (a finite sequence).

Original entry on oeis.org

1, 105263157894736842, 1034482758620689655172413793, 102564, 142857, 1016949152542372881355932203389830508474576271186440677966, 1014492753623188405797, 1012658227848, 10112359550561797752808988764044943820224719

Offset: 1

Lekraj Beedassy, Aug 21 2004; corrected Dec 17 2004

This is the least n-parasitic number. A k-parasitic number (where 1 <= k <= 9) is one such that when it is multiplied by k, the product obtained is merely its rightmost digit transferred in front at the leftmost end.

			102564 is 4-parasitic because we have 102564*4=410256.
For n=5: 142857*5=714285. [Dzmitry Paulenka (pavlenko(AT)tut.by), Aug 09 2009]

C. A. Pickover, Wonders of Numbers, Chapter 28, Oxford Univ. Press UK 2000.

Wikipedia, Parasitic numbers [From Dzmitry Paulenka (pavlenko(AT)tut.by), Aug 09 2009]
P. Yiu, k-left-transposable integers, Chap.18.2 pp. 168/360 in 'Recreational Mathematics'

Cf. A094676, A081463.

For other sequences with the same start, see A128857 and especially the cross-references in A097717.

Edited by N. J. A. Sloane, Apr 13 2009
Corrected to set 5th term to 142857 as this is the least 5-parasitic number. Dzmitry Paulenka (pavlenko(AT)tut.by), Aug 09 2009
a(9) added by Ian Duff, Jan 03 2012
Incorrect formula removed by Alois P. Heinz, Feb 18 2020

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

A128858 Number of digits in A128857(n).

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A097717 a(n) = least number m such that the quotient m/n is obtained merely by shifting the leftmost digit of m to the right end.

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A092697 For 1 <= n <= 9, a(n) = least number m such that the product n*m is obtained merely by shifting the rightmost digit of m to the left end (a finite sequence).

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

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