A333853 The values >= 2 of A135303 for the odd numbers A333855(n), for n >= 1.

2, 3, 2, 2, 3, 2, 2, 3, 4, 4, 4, 4, 3, 3, 2, 2, 2, 3, 4, 3, 2, 2, 9, 6, 3, 2, 4, 5, 2, 3, 3, 2, 2, 6, 2, 4, 2, 3, 2, 4, 2, 8, 2, 3, 6, 4, 4, 3, 3, 2, 4, 10, 3, 2, 5, 8, 16, 3, 4, 4, 6, 5, 3, 3, 4, 3, 2, 2, 2, 2

Offset: 1

Author

Wolfdieter Lang, Jun 29 2020

Keywords

• nonn

Comments

In Schick's book these are the B values, the number of periodic sequences, for the odd numbers N with B values >= 2. These numbers N are given in A333855.

In the complete coach system Sigma(b) of Hilton and Pedersen, these are the number of coaches for the odd numbers b from A333855 with more than one coach.

These are also the number of periodic modified doubling sequences for the odd numbers b from A333855 given in comments and examples by Gary W. Adamson, see his Aug 25 2019 comment in A065941, where this is named "r-t table" (for roots trajectory).

Examples

			n = 23: A333855(23) = 127 with A135303((127-1)/2) = A135303(63) = 9 = a(23). There are 9 Schick cycles (see also A333850), also 9 coaches, and also 9 modified doubling sequences.
		

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. A065941, A135303, A333850, A333855.

Programs

Mathematica

Map[EulerPhi[#2]/(2 If[#2 > 1 && GCD[#1, #2] == 1, Min[MultiplicativeOrder[#1, #2, {-1, 1}]], 0]) & @@ {2, #} &, 1 + 2 Select[Range[2, 15000], 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 *)

Formula

a(n) = A135303((A333855(n)-1)/2), for n >= 1.