A101202 Multiples of 142857.

142857, 285714, 428571, 571428, 714285, 857142, 999999, 1142856, 1285713, 1428570, 1571427, 1714284, 1857141, 1999998, 2142855, 2285712, 2428569, 2571426, 2714283, 2857140, 2999997, 3142854, 3285711, 3428568, 3571425, 3714282, 3857139, 3999996, 4142853, 4285710

Offset: 1

Author

Paul Boddington, Dec 12 2004

Keywords

• easy
• nonn

Comments

This sequence is interesting because the first six terms are cyclic shifts of one another (in decimal).

The number 142857 is the only number such that every cyclic permutation of it is a multiple of it. This number, its double, and numbers formed from either of them by an arbitrary number of repetitions of its sequence of digits are the only ones which generate more than one multiple of themselves in the sequence of their cyclic permutations. See the Kraitchik reference. - Martin Renner, Mar 06 2014

Actually, the second claim by Kraitchik is incorrect. Indeed, there are other numbers which generate more than one multiple of themselves, for example 153846 * {3, 4} = {461538, 615384} and 230769 * {3, 4} = {692307, 923076}. - Giovanni Resta, Mar 11 2014

The digits of each term can be partitioned into two parts with the right part always consisting of 3 digits and the left part containing all the remaining digits; adding then those two numbers will yield a sum which is multiple of 999: (142+857)/999 = 1 .. (857+142)/999 = 1 (999+999)/999 = 2 (1142+856)/999 = 2 .. (1857+141)/999 = 2 (2142+855)/999 = 3 .. (2857+140)/999 = 3 (2999+997)/999 = 4 .. As a result of above operations the sequence A054896 starting with the term 8 is produced. - Alexander R. Povolotsky, Nov 02 2007

1/7 = 0.142857142857142857... - Harvey P. Dale, Aug 26 2014

References

• Maurice Kraitchik, Mathematical Recreations, New York, Dover (2nd. ed.) 1953, p. 75-76.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 142857 at pp. 179-183.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A054896.

Programs

Mathematica

CoefficientList[Series[142857/(1 - x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 13 2014 *)
142857*Range[30] (* Harvey P. Dale, Aug 26 2014 *)

Formula

G.f.: 142857*x/(1-x)^2. - Alexander R. Povolotsky, Apr 26 2008

a(n) = 142857*n. - Robert Israel, May 14 2008

E.g.f.: 142857*x*exp(x). - Stefano Spezia, Sep 18 2024