A121153 - OEIS

A121153

Numbers n with the property that 1/n 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

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OFFSET

1,2

COMMENTS

Numbers n such that 1/n is in the Cantor set.

A subsequence of A054591. The first member of A054591 which does not belong to this sequence is 146. See A135666.

This is not a subsequence of A005836 (949 belongs to the present sequence but not to A005836). See A170830, A170853.

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]

EXAMPLE

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.

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] &]

PROG

(PARI) is(n, R=divrem(3^logint(n, 3), n), S=0)={while(R[1]!=1&&!bittest(S, R[2]), S+=1<<R[2]; R=divrem(R[2]*3, n)); R[1]!=1||R[2]==0} \\ M. F. Hasler, Feb 27 2018

CROSSREFS

Cf. A054591, A005823, A005836, A170943, A170944, A170951, A170952, A170830, A170853.

Sequence in context: A010439 A005836 A054591 * A276986 A283984 A283985

Adjacent sequences: A121150 A121151 A121152 * A121154 A121155 A121156

KEYWORD

nonn,base

AUTHOR

Jack W Grahl, Aug 12 2006

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

STATUS

approved