

A097933


Primes such that p divides 3^((p1)/2)  1.


9



11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167, 179, 181, 191, 193, 227, 229, 239, 241, 251, 263, 277, 311, 313, 337, 347, 349, 359, 373, 383, 397, 409, 419, 421, 431, 433, 443, 457, 467, 479, 491, 503, 541, 563, 577, 587, 599, 601, 613
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OFFSET

1,1


COMMENTS

Rational primes that decompose in the field Q[sqrt(3)].  N. J. A. Sloane, Dec 26 2017
For all primes p > 2 and integers gcd(x, y, p) = 1, x^((p1)/2) + y^((p1)/2) is divisible by p. This is because (x^((p1)/2)  y^((p1)/2))(x^((p1)/2) + y^((p1)/2)) = x^(p1)  y^(p1) is divisible by p according to Fermat's Little Theorem (FLT). This sequence lists p that divides 3^((p1)/2)  1^((p1)/2), and A003630 lists the '+' case.
Apart from initial terms, this and A038874 are the same.  N. J. A. Sloane, May 31 2009
Primes in A091998.  Reinhard Zumkeller, Jan 07 2012
Also, primes congruent to 1 or 11 (mod 12).  Vincenzo Librandi, Mar 23 2013
Conjecture: Let r(n) = (a(n)  1)/(a(n) + 1) if a(n) mod 4 = 1, (a(n) + 1)/(a(n)  1) otherwise; then Product_{n>=1} r(n) = (6/5) * (6/7) * (12/11) * (18/19) * ... = 2/sqrt(3).  Dimitris Valianatos, Mar 27 2017
Primes p such that Kronecker(12,p) = +1 (12 is the discriminant of Q[sqrt(3)]), that is, odd primes that have 3 as a quadratic residue.  Jianing Song, Nov 21 2018


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index to sequences related to decomposition of primes in quadratic fields


EXAMPLE

For p = 5, 3^2  1 = 8 <> 3*k for any integer k, so 5 is not in this sequence.
For p = 11, 3^5  1 = 242 = 11*22, so 11 is in this sequence.


MATHEMATICA

Select[Prime[Range[300]], MemberQ[{1, 11, 13, 23}, Mod[#, 24]]&] (* Vincenzo Librandi, Mar 23 2013 *)


PROG

(PARI) /* s = +1, d=diff */ ptopm1d2(n, x, d, s) = { forprime(p=3, n, p2=(p1)/2; y=x^p2 + s*(xd)^p2; if(y%p==0, print1(p", "))) }
(PARI) {a(n)= local(m, c); if(n<1, 0, c=0; m=0; while( c<n, m++; if( isprime(m)& kronecker(3, m)==1, c++)); m)} /* Michael Somos, Aug 28 2006 */
(Haskell)
a097933 n = a097933_list !! (n1)
a097933_list = [x  x < a091998_list, a010051 x == 1]
 Reinhard Zumkeller, Jan 07 2012
(MAGMA) [p: p in PrimesUpTo(1000)  p mod 24 in [1, 11, 13, 23]]; // Vincenzo Librandi, Mar 23 2013


CROSSREFS

Cf. A003630, A010051, A038874, A091998.
Sequence in context: A136058 A106073 A072330 * A166484 A127043 A084952
Adjacent sequences: A097930 A097931 A097932 * A097934 A097935 A097936


KEYWORD

nonn


AUTHOR

Cino Hilliard, Sep 04 2004


STATUS

approved



