%I #20 Oct 15 2020 16:34:53
%S 17,31,33,41,43,51,57,63,65,73,85,89,91,93,97,99,105,109,113,117,119,
%T 123,127,129,133,137,145,151,153,155,157,161,165,171,177,185,187,189,
%U 193,195,201,205,209,215,217,219,221,223,229,231,233,241,247,249,251,255
%N Numbers 2*k + 1 with A135303(k) >= 2, for k >= 1, sorted increasingly.
%C These are the numbers a(n) for which there is more than one periodic Schick sequence. In his notation B(a(n)) >= 2, for n >= 1.
%C These are also the numbers a(n) for which there is more than one coach in the complete coach system Sigma(b = a(n)) of Hilton and Pedersen, for n >= 1
%C These are the numbers a(n) for which there is more than one cycle in the complete system MDS(a(n)) (Modified Doubling Sequence) proposed in the comment by _Gary W. Adamson_, Aug 20 2019, in A003558.
%C The complement relative to the odd numbers >= 3 is given in A333854.
%C The subsequence for odd primes is identical with A268923.
%D Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, pp. 261-264.
%D Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Bokos Druck, Zürich, 2003 (ISBN 3-9522917-0-6). Tables 3.1 to 3.10, for odd p = 3..113 (with gaps), pp. 158-166.
%H Michael De Vlieger, <a href="/A333855/b333855.txt">Table of n, a(n) for n = 1..10000</a>
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/2008.04300">On the Equivalence of Three Complete Cyclic Systems of Integers</a>, arXiv:2008.04300 [math.NT], 2020
%F Sequence {a(n)}_{n>=1} of numbers 2*k + 1 satisfying A135303(k) >= 2, for k >= 1, ordered increasingly.
%t 1 + 2 Select[Range[2, 127], 2 <= EulerPhi[#2]/(2 If[#2 > 1 && GCD[#1, #2] == 1, Min[MultiplicativeOrder[#1, #2, {-1, 1}]], 0]) & @@ {2, 2 # + 1} &] (* _Michael De Vlieger_, Oct 15 2020 *)
%o (PARI) isok8(m, n) = my(md = Mod(2, 2*n+1)^m); (md==1) || (md==-1);
%o A003558(n) = my(m=1); while(!isok8(m, n) , m++); m;
%o isok(m) = (m%2) && eulerphi(m)/(2*A003558((m-1)/2)) >= 2; \\ _Michel Marcus_, Jun 09 2020
%Y Cf. A003558, A135303, A216371, A268923, A333854.
%K nonn
%O 1,1
%A _Wolfdieter Lang_, May 12 2020