A121153 Numbers k with the property that 1/k can be written in base 3 in such a way that the fractional part contains no 1's.
1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 36, 39, 40, 81, 82, 84, 90, 91, 108, 117, 120, 121, 243, 244, 246, 252, 270, 273, 324, 328, 351, 360, 363, 364, 729, 730, 732, 738, 756, 757, 810, 819, 820, 949, 972, 984, 1036, 1053, 1080, 1089, 1092, 1093, 2187
Offset: 1
Examples
1/3 in base 3 can be written as either .1 or .0222222... The latter version contains no 1's, so 3 is in the sequence. 1/4 in base 3 is .02020202020..., so 4 is in the sequence.
Links
- T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n=1..1164 (terms < 3^21)
- D. Jordan and R. Schayer Rational points on the Cantor middle thirds set [Broken link corrected by _Rainer Rosenthal_, Feb 20 2009]
Programs
-
Mathematica
(* Mma code from T. D. Noe, Feb 20 2010. This produces the sequence except for the powers of 3. *) (* Find the length of the periodic part of the fraction: *) FracLen[n_] := Module[{r = n/3^IntegerExponent[n, 3]}, MultiplicativeOrder[3, r]] (* Generate the fractions and select those that have no 1's: *) Select[Range[100000], ! MemberQ[Union[RealDigits[1/#, 3, FracLen[ # ]][[1]]], 1] &]
-
PARI
is(n,R=divrem(3^logint(n,3),n),S=0)={while(R[1]!=1&&!bittest(S,R[2]), S+=1<
Extensions
Extended to 10^5 by T. D. Noe and N. J. A. Sloane, Feb 20 2010
Entry revised by N. J. A. Sloane, Feb 22 2010
Comments