login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A296920
Rational primes that decompose in the quadratic field Q(sqrt(-11)).
29
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
OFFSET
1,1
COMMENTS
Primes that are 1, 3, 5, 9, or 15 mod 22. - Charles R Greathouse IV, Mar 18 2018
(Which means: union of A141849, A141850, A141852, A141856 and A141851. - R. J. Mathar, Apr 15 2024)
REFERENCES
Helmut Hasse, Number Theory, Grundlehren 229, Springer, 1980, page 498.
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");
end if;
q:=-D;
SD:=[]; SI:=[]; QR:=[]; NR:=[];
if (D mod 8) = 1 then
SD:=[op(SD), 2];
end if;
if (D mod 8) = 5 then
SI:=[op(SI), 2];
end if;
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];
end if;
if (D mod p) <> 0 and legendre(D, p)=-1 then
SI:=[op(SI), p];
end if;
end do;
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];
end if;
end do:
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);
end if;
end proc:
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
Sequence in context: A222424 A067256 A136891 * A106857 A106307 A106282
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 25 2017
STATUS
approved