A284601 Numbers k such that the decimal representation of 1/k does not terminate and has odd period.

3, 6, 9, 12, 15, 18, 24, 27, 30, 31, 36, 37, 41, 43, 45, 48, 53, 54, 60, 62, 67, 71, 72, 74, 75, 79, 81, 82, 83, 86, 90, 93, 96, 106, 107, 108, 111, 120, 123, 124, 129, 134, 135, 142, 144, 148, 150, 151, 155, 158, 159, 162, 163, 164, 166, 172, 173, 180, 185, 186, 191, 192, 199, 201, 205, 212, 213, 214, 215

Offset: 1

Author

Ilya Gutkovskiy, Mar 30 2017

Keywords

• nonn
• base

Comments

From Robert G. Wilson v, Apr 02 2017: (Start)

If k is in the sequence, then so are 2k and 5k.

The complement of A284602.

Primitives: 3, 9, 27, 31, 37, 41, 43, 53, 67, 71, 79, 81, 83, 93, 107, 111, 123, ..., .

(End)

From Robert Israel, Apr 03 2017: (Start)

Numbers of the form 2^j * 5^k * m where m > 1, gcd(m,10)=1 and the multiplicative order of 10 (mod m) is odd.

Complement of A003592 in the multiplicative semigroup generated by A186635, i.e., numbers whose prime factors are in A186635 with at least one prime factor not 2 or 5. (End)

Examples

			27 is in the sequence because 1/27 = 0.0370(370)... period is 3, 3 is odd.
2 and 5 are not in the sequence because 1/2 = 0.5 and 1/5 = 0.2 are terminating expansions. See also comments in A051626 and A284602.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

Cf. A002371, A003592, A003814, A051626, A186635, A284602.

Programs

Maple

filter:= proc(n) local m;
  m:= n/2^padic:-ordp(n,2);
  m:= m/5^padic:-ordp(m,5);
  m > 1 and numtheory:-order(10,m)::odd
end proc:
select(filter, [$1..1000]); # Robert Israel, Apr 03 2017

Mathematica

Select[Range[215], Mod[Length[RealDigits[1/#][[1, -1]]], 2] == 1 & ]