

A216371


Odd primes with one coach: primes p such that A135303((p1)/2) = 1.


6



3, 5, 7, 11, 13, 19, 23, 29, 37, 47, 53, 59, 61, 67, 71, 79, 83, 101, 103, 107, 131, 139, 149, 163, 167, 173, 179, 181, 191, 197, 199, 211, 227, 239, 263, 269, 271, 293, 311, 317, 347, 349, 359, 367, 373, 379, 383, 389, 419, 421, 443, 461, 463, 467, 479, 487
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OFFSET

1,1


COMMENTS

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.
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


REFERENCES

P. Hilton and J. Pedersen, A Mathematical Tapestry, Demonstrating the Beautiful Unity of Mathematics, 2010, Cambridge University Press, pages 260264.


LINKS

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)


FORMULA

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
a(n) = 2*A054639(n) + 1.  L. Edson Jeffery, Dec 18 2012


EXAMPLE

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.
13 is in the set since A135303(6) = 1; but 17 isn't since A135303(8) = 2.


MAPLE

isA216371 := proc(n)
if isprime(n) then
if A135303((n1)/2) = 1 then
true;
else
false;
end if;
else
false;
end if;
end proc:
A216371 := proc(n)
local p;
if n = 1 then
3;
else
p := nextprime(procname(n1)) ;
while true do
if isA216371(p) then
return p;
end if;
p := nextprime(p) ;
end do:
end if;
end proc:
seq(A216371(n), n=1..40) ; # R. J. Mathar, Dec 01 2014


MATHEMATICA

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 *)
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 *)


PROG

(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


CROSSREFS

Cf. A003558, A135303, A000040.
Cf. A054639.
Sequence in context: A095070 A079733 A179538 * A095747 A132779 A192869
Adjacent sequences: A216368 A216369 A216370 * A216372 A216373 A216374


KEYWORD

nonn


AUTHOR

Gary W. Adamson, Sep 05 2012


STATUS

approved



