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

%I

%S 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,

%T 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,

%U 2,2,2

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

%C 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.

%C 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.

%C 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. Admason_, See his Aug 25 2019 comment in A065941, where this is named "r-t table" (for roots trajectory).

%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="/A333853/b333853.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 a(n) = A135303((A333855(n)-1)/2), for n >= 1.

%e 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.

%t 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 *)

%Y Cf. A065941, A135303, A333850, A333855.

%K nonn

%O 1,1

%A _Wolfdieter Lang_, Jun 29 2020

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Last modified September 20 06:32 EDT 2021. Contains 347577 sequences. (Running on oeis4.)