A180340 Numbers with x digits such that the first x multiples are cyclic permutations of the number, leading 0's omitted (or cyclic numbers).

142857, 588235294117647, 52631578947368421, 434782608695652173913, 344827586206896551724137931, 212765957446808510638297872340425531914893617, 169491525423728813559322033898305084745762711864406779661

Offset: 1

Author

Ralph Kerchner (daxkerchner(AT)hotmail.com), Aug 28 2010

Keywords

• base
• nonn

Comments

Periodic part of decimal expansion of 1/A001913(n). The number of digits in each term (including leading zeros), plus one, makes the sequence A001913.

Examples

			142857 is in the sequence because it has 6 digits and the first 6 multiples of 142857 are 142857, 285714, 428571, 571428, 714285, and 857142, all cyclic permutations of the number. Also the first term of A001913 is 7, and 1/7 = 0.142857142857... .
588235294117647 is the next number because 0588235294117647 has 16 digits and the first 16 multiples are cyclic permutations of the number; the second term of A001913 is 17, and 1/17 = 0.05882352941176470588235294117647... .
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..60
• Edwin E. Freed, Binary Magic Numbers, Dr. Dobb's Journal, Vol. 78 (April 1983), pp. 24-37.
• OEIS Wiki, Cyclic numbers
• Eric Weisstein's World of Mathematics, Cyclic number
• Wikipedia, Cyclic number

Crossrefs

A006883 starting from the second term of A006883, omitting ending 0's.

The n-th terms of A060284 where n is a member of A001913.

Programs

Mathematica

Map[(10^(# - 1) - 1)/# &, Select[Prime@ Range@ 17, MultiplicativeOrder[10, #] == # - 1 &]] (* Michael De Vlieger, Apr 03 2017 *)

Formula

a(n) = (10^(A001913(n)-1) - 1) / A001913(n).