|
|
A141163
|
|
Primes of the form x^2+12*x*y-y^2.
|
|
7
|
|
|
37, 67, 107, 137, 139, 151, 233, 269, 293, 317, 349, 367, 491, 601, 691, 823, 839, 863, 877, 881, 929, 941, 971, 1061, 1069, 1103, 1163, 1237, 1259, 1279, 1283, 1307, 1373, 1433, 1489, 1553, 1601, 1607, 1627, 1669, 1693, 1777, 1783, 1787, 1847, 1877, 1973
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Discriminant = 148. Class = 3. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.
|
|
REFERENCES
|
Z. I. Borevich and I. R. Shafarevich, Number Theory.
|
|
LINKS
|
|
|
EXAMPLE
|
a(4)=137 because we can write 137= 3^2+12*3*4-4^2.
|
|
MATHEMATICA
|
q := x^2 + 12*x*y - y^2; pmax = 2000; xmax = 100; ymin = -xmax; ymax = xmax; k = 1; prms0 = {}; prms = {2}; While[prms != prms0, xx = yy = {}; prms0 = prms; prms = Reap[Do[p = q; If[2 <= p <= pmax && PrimeQ[p], AppendTo[xx, x]; AppendTo[yy, y]; Sow[p]], {x, 1, k*xmax}, {y, k *ymin, k *ymax}]][[2, 1]] // Union; xmax = Max[xx]; ymin = Min[yy]; ymax = Max[yy]; k++; Print["k = ", k, " xmax = ", xmax, " ymin = ", ymin, " ymax = ", ymax ]]; A141163 = prms (* Jean-François Alcover, Oct 26 2016 *)
|
|
PROG
|
(Sage)
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 12, -1])
print(Q.represented_positives(1973, 'prime')) # Peter Luschny, Oct 26 2016
|
|
CROSSREFS
|
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jun 13 2008
|
|
STATUS
|
approved
|
|
|
|