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A216371 Odd primes with one coach: primes p such that A135303((p-1)/2) = 1. 6

%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((p-1)/2) = 1.

%C Given that prime p has only one coach, the corresponding value of k in A003558 must be (p-1)/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) = (p-1), and k must be (p-1)/2 iff c = 1. Or, phi(p) = (p-1) = 2 * 1 * (p-1)/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 260-264.

%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 < (p-1)/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((n-1)/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(n-1)) ;

%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\2-1, 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

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Last modified March 21 07:23 EDT 2019. Contains 321367 sequences. (Running on oeis4.)