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

%I A284601 #38 Nov 19 2023 21:17:15
%S A284601 3,6,9,12,15,18,24,27,30,31,36,37,41,43,45,48,53,54,60,62,67,71,72,74,
%T A284601 75,79,81,82,83,86,90,93,96,106,107,108,111,120,123,124,129,134,135,
%U A284601 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
%N A284601 Numbers k such that the decimal representation of 1/k does not terminate and has odd period.
%C A284601 From _Robert G. Wilson v_, Apr 02 2017: (Start)
%C A284601 If k is in the sequence, then so are 2k and 5k.
%C A284601 The complement of A284602.
%C A284601 Primitives: 3, 9, 27, 31, 37, 41, 43, 53, 67, 71, 79, 81, 83, 93, 107, 111, 123, ..., .
%C A284601 (End)
%C A284601 From _Robert Israel_, Apr 03 2017: (Start)
%C A284601 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.
%C A284601 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)
%H A284601 Robert G. Wilson v, <a href="/A284601/b284601.txt">Table of n, a(n) for n = 1..10000</a>
%H A284601 <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>
%e A284601 27 is in the sequence because 1/27 = 0.0370(370)... period is 3, 3 is odd.
%e A284601 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.
%p A284601 filter:= proc(n) local m;
%p A284601   m:= n/2^padic:-ordp(n,2);
%p A284601   m:= m/5^padic:-ordp(m,5);
%p A284601   m > 1 and numtheory:-order(10,m)::odd
%p A284601 end proc:
%p A284601 select(filter, [$1..1000]); # _Robert Israel_, Apr 03 2017
%t A284601 Select[Range[215], Mod[Length[RealDigits[1/#][[1, -1]]], 2] == 1 & ]
%Y A284601 Cf. A002371, A003592, A003814, A051626, A186635, A284602.
%K A284601 nonn,base
%O A284601 1,1
%A A284601 _Ilya Gutkovskiy_, Mar 30 2017