A333855 Numbers 2*k + 1 with A135303(k) >= 2, for k >= 1, sorted increasingly.

17, 31, 33, 41, 43, 51, 57, 63, 65, 73, 85, 89, 91, 93, 97, 99, 105, 109, 113, 117, 119, 123, 127, 129, 133, 137, 145, 151, 153, 155, 157, 161, 165, 171, 177, 185, 187, 189, 193, 195, 201, 205, 209, 215, 217, 219, 221, 223, 229, 231, 233, 241, 247, 249, 251, 255

Offset: 1

Author

Wolfdieter Lang, May 12 2020

Keywords

• nonn

Comments

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.

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

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.

The complement relative to the odd numbers >= 3 is given in A333854.

The subsequence for odd primes is identical with A268923.

References

• Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, pp. 261-264.
• 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.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020

Crossrefs

Cf. A003558, A135303, A216371, A268923, A333854.

Programs

Mathematica

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 *)

PARI

isok8(m, n) = my(md = Mod(2, 2*n+1)^m); (md==1) || (md==-1);
A003558(n) = my(m=1); while(!isok8(m, n) , m++); m;
isok(m) = (m%2) && eulerphi(m)/(2*A003558((m-1)/2)) >= 2; \\ Michel Marcus, Jun 09 2020

Formula

Sequence {a(n)}_{n>=1} of numbers 2*k + 1 satisfying A135303(k) >= 2, for k >= 1, ordered increasingly.