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

1,2

Comments

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

Links

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.

Example

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).

Mathematica

(*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}]

Prog

(Python)
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

Crossrefs

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

Sequence in context: A217592 A092697 A097717 * A357515 A246111 A067818

Adjacent sequences: A128854 A128855 A128856 * A128858 A128859 A128860

Keyword

nonn,base

Author

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

Extensions

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

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

Status

approved