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 A296920 Rational primes that decompose in the quadratic field Q(sqrt(-11)). 27
 3, 5, 23, 31, 37, 47, 53, 59, 67, 71, 89, 97, 103, 113, 137, 157, 163, 179, 181, 191, 199, 223, 229, 251, 257, 269, 311, 313, 317, 331, 353, 367, 379, 383, 389, 397, 401, 419, 421, 433, 443, 449, 463, 467, 487, 499, 509, 521, 577, 587, 599, 617, 619, 631, 641, 643, 647, 653, 661, 683, 691, 709, 719 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Primes that are 1, 3, 5, 9, or 15 mod 22. - Charles R Greathouse IV, Mar 18 2018 REFERENCES Helmut Hasse, Number Theory, Grundlehren 229, Springer, 1980, page 498. LINKS Robert Israel, Table of n, a(n) for n = 1..10000 FORMULA a(n) ~ 2n log n. - Charles R Greathouse IV, Mar 18 2018 MAPLE # In the quadratic field Q(sqrt(D)), for squarefree D<0, compute lists of: # rational primes that decompose (SD), # rational primes that are inert (SI), # primes p such that D is a square mod p (QR), and # primes p such that D is a nonsquare mod p (NR), # omitting the latter if it is the same as the inert primes. # Consider first M primes p. # Reference: Helmut Hasse, Number Theory, Grundlehren 229, Springer, 1980, page 498. with(numtheory): HH := proc(D, M) local SD, SI, QR, NR, p, q, i, t1; # if D >= 0 then error("D must be negative"); fi; if not issqrfree(D) then error("D must be squarefree"); fi; q:=-D; SD:=[]; SI:=[]; QR:=[]; NR:=[]; if (D mod 8) = 1 then SD:=[op(SD), 2]; fi; if (D mod 8) = 5 then SI:=[op(SI), 2]; fi; for i from 2 to M do p:=ithprime(i); if (D mod p) <> 0 and legendre(D, p)=1 then SD:=[op(SD), p]; fi; if (D mod p) <> 0 and legendre(D, p)=-1 then SI:=[op(SI), p]; fi; od; for i from 1 to M do p:=ithprime(i); if legendre(D, p) >= 0 then QR:=[op(QR), p]; else NR:=[op(NR), p]; fi; od: lprint("Primes that decompose:", SD); lprint("Inert primes:", SI); lprint("Primes p such that Legendre(D, p) = 0 or 1: ", QR); if SI <> NR then lprint("Note: SI <> NR here!"); lprint("Primes p such that Legendre(D, p) = -1: ", NR); fi; end; HH(-11, 200); produces the present sequence (A296920), A191060, and A056874. MATHEMATICA Reap[For[p = 2, p < 1000, p = NextPrime[p], If[KroneckerSymbol[-11, p] == 1, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Apr 29 2019 *) PROG (PARI) list(lim)=my(v=List()); forprime(p=2, lim, if(kronecker(-11, p)==1, listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Mar 18 2018 CROSSREFS Cf. A191060, A056874. Sequence in context: A222424 A067256 A136891 * A106857 A106307 A106282 Adjacent sequences:  A296917 A296918 A296919 * A296921 A296922 A296923 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Dec 25 2017 STATUS approved

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Last modified December 9 19:51 EST 2019. Contains 329879 sequences. (Running on oeis4.)