|
%I M3764 #62 Feb 24 2023 15:11:36
%S 5,7,13,23,29,31,37,47,53,61,71,79,101,103,109,127,149,151,157,167,
%T 173,181,191,197,199,223,229,239,263,269,271,277,293,311,317,349,359,
%U 367,373,383,389,397,421,431,439
%N Primes congruent to {5, 7} mod 8.
%C Inert rational odd primes in the field Q(sqrt(-2)).
%C Primes p such that p XOR 5 = p - 5. - _Brad Clardy_, Jul 22 2012
%C Terms m in A047566 with A010051(m) = 1. - _Reinhard Zumkeller_, Dec 29 2012
%C This sequence gives the primes p which satisfy norm(rho(p)) = - 1 with rho(p) := 2*cos(Pi/p) (the length ratio (smallest diagonal)/side in the regular p-gon). The norm of an algebraic number (over Q) is the product over all zeros of its minimal polynomial. Here norm(rho(p)) = (-1)^delta(p)* C(p, 0), with the degree delta(p) = A055034(p) = (p-1)/2. For p == 5 (mod 8) the norm is C(p, 0) (see a comment on 2*A230076) and for p == 7 (mod 8) the norm is -C(p, 0) (see a comment on A186302). For the primes with norm(rho(p)) = +1 see A033200. - _Wolfdieter Lang_, Oct 24 2013
%D H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 498.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Reinhard Zumkeller, <a href="/A003628/b003628.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="https://oeis.org/index/Pri#primes_decomp_of">Index to sequences related to decomposition of primes in quadratic fields</a>
%F a(n) ~ 2n log n. - _Charles R Greathouse IV_, Feb 24 2023
%t Select[Prime[Range[200]],MemberQ[{5,7},Mod[#,8]]&] (* _Harvey P. Dale_, Oct 24 2011 *)
%o (Haskell)
%o a003628 n = a003628_list !! (n-1)
%o a003628_list = filter ((== 1) . a010051) a047566_list
%o -- _Reinhard Zumkeller_, Dec 29 2012, Jan 22 2012
%o (PARI) {a(n) = local( cnt, m ); if( n<1, return( 0 )); while( cnt < n, if( isprime( m++) && kronecker( -2, m )==-1, cnt++ )); m} /* _Michael Somos_, Aug 14 2012 */
%o (Magma) [ p: p in PrimesUpTo(600) | p mod 8 in {5, 7}]; // _Vincenzo Librandi_, Aug 22 2012
%Y Cf. A000040, A039706, A033203 (complement with respect to A000040).
%K nonn,easy,nice
%O 1,1
%A _N. J. A. Sloane_, _Mira Bernstein_
|
