%I
%S 3,5,7,11,13,19,23,29,37,47,53,59,61,67,71,79,83,101,103,107,131,139,
%T 149,163,167,173,179,181,191,197,199,211,227,239,263,269,271,293,311,
%U 317,347,349,359,367,373,379,383,389,419,421,443,461,463,467,479,487
%N Odd primes with one coach: primes p such that A135303((p1)/2) = 1.
%C Given that prime p has only one coach, the corresponding value of k in A003558 must be (p1)/2, and vice versa. Using the Coach theorem of Jean Pedersen et al., phi(b) = 2 * c * k, with b odd. Let b = p, prime. Then phi(p) = (p1), and k must be (p1)/2 iff c = 1. Or, phi(p) = (p1) = 2 * 1 * (p1)/2.
%C Conjecture relating to odd integers: iff an integer is in the set A216371 and is either of the form 4q  1 or 4q + 1, (q>0); then the top row of its coach (Cf. A003558) is composed of a permutation of the first q odd integers. Examples: 11 is of the form 4q  1, q = 3; with the top row of its coach [1, 5, 3]. 13 is of the form 4q + 1, q = 3; so has a coach of [1, 3, 5]. 37 is of the form 4q + 1, q = 9; so has a coach with the top row composed of a permutation of the first 9 odd integers: [1, 9, 7, 15, 11, 13, 3, 17, 5].  _Gary W. Adamson_, Sep 08 2012
%D P. Hilton and J. Pedersen, A Mathematical Tapestry, Demonstrating the Beautiful Unity of Mathematics, 2010, Cambridge University Press, pages 260264.
%H T. D. Noe and Charles R Greathouse IV, <a href="/A216371/b216371.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%F Odd primes p such that 2^m is not 1 or 1 mod p for 0 < m < (p1)/2.  _Charles R Greathouse IV_, Sep 15 2012
%F a(n) = 2*A054639(n) + 1.  _L. Edson Jeffery_, Dec 18 2012
%e Prime 23 has a k value of 11 = (23  1)/2 (Cf. A003558(11). It follows that 23 has only one coach (A135303(11) = 1). 23 is thus in the set. On the other hand 31 is not in the set since A135303(15) shows 3 coaches, with A003558(15) = 5.
%e 13 is in the set since A135303(6) = 1; but 17 isn't since A135303(8) = 2.
%p isA216371 := proc(n)
%p if isprime(n) then
%p if A135303((n1)/2) = 1 then
%p true;
%p else
%p false;
%p end if;
%p else
%p false;
%p end if;
%p end proc:
%p A216371 := proc(n)
%p local p;
%p if n = 1 then
%p 3;
%p else
%p p := nextprime(procname(n1)) ;
%p while true do
%p if isA216371(p) then
%p return p;
%p end if;
%p p := nextprime(p) ;
%p end do:
%p end if;
%p end proc:
%p seq(A216371(n),n=1..40) ; # _R. J. Mathar_, Dec 01 2014
%t Suborder[a_, n_] := If[n > 1 && GCD[a, n] == 1, Min[MultiplicativeOrder[a, n, {1, 1}]], 0]; nn = 150; Select[Prime[Range[2, nn]], EulerPhi[#]/(2*Suborder[2, #]) == 1 &] (* _T. D. Noe_, Sep 18 2012 *)
%t f[p_] := Sum[Cos[2^n Pi/((2 p + 1))], {n, p}]; 1 + 2 * Select[Range[500], Reduce[f[#] == 1/2, Rationals] &]; (* _Gerry Martens_, May 01 2016 *)
%o (PARI) is(p)=for(m=1,p\21, if(abs(centerlift(Mod(2,p)^m))==1, return(0))); p>2 && isprime(p) \\ _Charles R Greathouse IV_, Sep 18 2012
%Y Cf. A003558, A135303, A000040.
%Y Cf. A054639.
%K nonn
%O 1,1
%A _Gary W. Adamson_, Sep 05 2012
