A333854 Numbers 2*k + 1 with A135303(k) = 1, for k >= 1, sorted increasingly.

3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 35, 37, 39, 45, 47, 49, 53, 55, 59, 61, 67, 69, 71, 75, 77, 79, 81, 83, 87, 95, 101, 103, 107, 111, 115, 121, 125, 131, 135, 139, 141, 143, 147, 149, 159, 163, 167, 169, 173, 175, 179, 181, 183, 191, 197, 199, 203

Offset: 1

Author

Wolfdieter Lang, May 03 2020

Keywords

• nonn

Comments

These are the numbers a(n) for which there is only one periodic Schick sequence. In Schick's notation B(a(n)) = 1, for n >= 1.

These are the numbers a(n) for which there is only one coach in the complete coach system Sigma(b = a(n)) of Hilton and Pedersen, for n >= 1.

These are also the numbers a(n) for which there is only 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 subsequence of prime numbers is A216371.

The complement relative to the odd numbers >= 3 is given in A333855.

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.

Crossrefs

Cf. A003558, A135303, A216371, A268923, A333855.

Programs

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)) == 1; \\ Michel Marcus, Jun 10 2020

Formula

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