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

1,1

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.

Links

Table of n, a(n) for n=1..60.

Formula

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

Prog

(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

Crossrefs

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

Sequence in context: A005842 A204458 A192861 * A192868 A283553 A081110

Adjacent sequences: A333851 A333852 A333853 * A333855 A333856 A333857

Keyword

nonn

Author

Wolfdieter Lang, May 03 2020

Status

approved