A038883 -id:A038883 - cp's OEIS frontend

A141165 Primes of the form 9x^2+7xy-5y^2.

3, 5, 11, 17, 19, 43, 61, 71, 83, 97, 103, 149, 151, 167, 181, 233, 271, 277, 293, 307, 311, 337, 367, 373, 397, 401, 409, 421, 431, 433, 457, 463, 467, 491, 557, 569, 587, 631, 641, 661, 673, 683, 701, 733, 743, 751, 757, 769, 787, 821, 859, 863, 883, 911

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jun 12 2008

nonn

Discriminant = 229. Class = 3. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac. They can represent primes only if gcd(a,b,c)=1. [Edited by M. F. Hasler, Jan 27 2016]
Also primes represented by the improperly equivalent form 5*x^2+7*x*y-9*y^2. - Juan Arias-de-Reyna, Mar 17 2011
36*a(n) has the form z^2 - 229*y^2, where z = 18*x+7*y. [Bruno Berselli, Jun 25 2014]
Appears to be the complement of A141166 in A268155, primes that are squares mod 229. - M. F. Hasler, Jan 27 2016

			a(10)=97 because we can write 97= 9*3^2+7*3*1-5*1^2

Z. I. Borevich and I. R. Shafarevich, Number Theory
D. B. Zagier, Zetafunktionen und quadratische Körper

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
Peter Luschny, Binary Quadratic Forms
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141166 (d=229).

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

q := 9*x^2 + 7*x*y - 5*y^2; pmax = 1000; xmax = xmax0 = 50; ymin = ymin0 = -50; ymax = ymax0 = 50; k = 1.3 (expansion coeff. for maxima *); 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, If[xmax == xmax0, xmax, Floor[k*xmax]]}, {y, If[ymin == ymin0, ymin, Floor[k*ymin]], If[ymax == ymax0, ymax, Floor[k*ymax]]}]][[2, 1]] // Union; xmax = Max[xx]; ymin = Min[yy]; ymax = Max[yy]; Print[Length[prms], " terms", "  xmax = ", xmax, "  ymin = ", ymin, "  ymax = ", ymax ]]; A141165 = prms (* Jean-François Alcover, Oct 26 2016 *)

is_A141165(p)=qfbsolve(Qfb(9,7,-5),p) \\ Returns nonzero (actually, a solution [x,y]) iff p is a member of the sequence. For efficiency it is assumed that p is prime. - M. F. Hasler, Jan 27 2016

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([9, 7, -5])
print(Q.represented_positives(911, 'prime')) # Peter Luschny, Oct 26 2016

A141166 Primes of the form x^2+15xy-y^2.

Original entry on oeis.org

37, 53, 173, 193, 229, 241, 347, 359, 383, 439, 443, 449, 461, 503, 509, 541, 593, 607, 617, 619, 643, 691, 907, 967, 977, 1019, 1051, 1063, 1097, 1109, 1249, 1277, 1291, 1303, 1321, 1399, 1429, 1583, 1667, 1741, 1783, 1993, 1997, 2003, 2087, 2137, 2143, 2333, 2347, 2351

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jun 12 2008

nonn

Discriminant = 229. Class = 3. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d = b^2-4ac. They can represent primes only if gcd(a,b,c)=1. [Edited by M. F. Hasler, Jan 27 2016]

Appears to be the complement of A141165 in A268155, primes that are squares mod 229. - M. F. Hasler, Jan 27 2016

			a(2)=53 because we can write 53= 3^2+15*3*1-1^2

Z. I. Borevich and I. R. Shafarevich, Number Theory

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141165 (d=229).

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

lim = 100; Rest@ Union@ Abs@ Flatten@ Table[x^2 + 15 x y - y^2, {x, lim}, {y, lim}] /. n_ /; CompositeQ@ n -> Nothing (* Michael De Vlieger, Jan 27 2016 *)

is_A141166(p)=qfbsolve(Qfb(1,15,-1),p) \\ Returns nonzero (actually, a solution [x,y]) iff p is a member of the sequence. For efficiency it is assumed that p is prime. Example usage: select(is_A141166,primes(500)) - M. F. Hasler, Jan 27 2016

A141171 Primes of the form -x^2+4xy+2y^2 (as well as of the form 5x^2+8xy+2*y^2).

Original entry on oeis.org

2, 5, 23, 29, 47, 53, 71, 101, 149, 167, 173, 191, 197, 239, 263, 269, 293, 311, 317, 359, 383, 389, 431, 461, 479, 503, 509, 557, 599, 647, 653, 677, 701, 719, 743, 773, 797, 821, 839, 863, 887, 911, 941, 983, 1013, 1031, 1061, 1103, 1109, 1151, 1181, 1223, 1229, 1277, 1301, 1319, 1367

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 12 2008

nonn

Discriminant is 24. Class is 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.
Also primes of form 6*u^2 - v^2. The transformation {u, v} = {y, x - 2*y} yields the form in the title. - Juan Arias-de-Reyna, Mar 19 2011
Members of A141171 but not of A105880: 2, 431, 911, 1013, 1181, ..., . - Robert G. Wilson v, Aug 30 2013
This is also the list of primes p such that p = 2 or p is congruent to 5 or 23 mod 24 - Jean-François Alcover, Oct 28 2016

			a(4) = 29 because we can write 29 = -1^2 + 4*1*3 + 2*3^2 (or 29 = 5*1^2 + 8*1*2 + 2*2^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.
D. B. Zagier, Zetafunktionen und quadratische Körper.

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A141170 (d = 24), A105880 (Primes for which -8 is a primitive root.) A038872 (d = 5). A038873 (d = 8). A068228, A141123 (d = 12). A038883 (d = 13). A038889 (d = 17). A141111, A141112 (d = 65).
Cf. also A242665.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

N:= 10^5: # to get all terms <= N
select(t -> isprime(t) and [isolve(6*u^2-v^2=t)]<>[], [2, seq(op([24*i+5,24*i+23]),i=0..floor((N-5)/24))]); # Robert Israel, Sep 28 2014

A141171 = {}; Do[p = -x^2 + 4 * x * y + 2 * y^2; If[p > 0 && PrimeQ@ p, AppendTo[A141171, p]], {x, 25}, {y, 25}]; Take[ Union@ A141171, 57] (* Robert G. Wilson v, Aug 30 2013 *)
Select[Prime[Range[250]], # == 2 || MatchQ[Mod[#, 24], 5|23]&] (* Jean-François Alcover, Oct 28 2016 *)

A141183 Primes of the form -x^2+6xy+2y^2 (as well as of the form 7x^2+10xy+2*y^2).

Original entry on oeis.org

2, 7, 11, 19, 43, 79, 83, 107, 127, 131, 139, 151, 167, 211, 227, 239, 263, 271, 283, 307, 347, 359, 431, 439, 479, 491, 503, 523, 547, 563, 571, 607, 659, 739, 743, 787, 811, 827, 887, 919, 967, 1019, 1031, 1051, 1063, 1091, 1151, 1163, 1187, 1223, 1231, 1283, 1319, 1327

Offset: 1

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

nonn

Discriminant = 44. Class = 2. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.
Also primes of form 11*u^2-v^2. The transformation {u,v}={-3*x-y,10*x+3*y} yields the form in the title. - Juan Arias-de-Reyna, Mar 20 2011
Also primes p equal -1 mod 4 and = 1, 3, 4, 5, or 9  mod 11. - Juan Arias-de-Reyna, Mar 20 2011

			a(4)=19 because we can write 19= -1^2+6*1*2+2*2^2 (or 19=7*1^2+10*1*1+2*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, NY, 1966.

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141182 (d=44).

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Select[Prime[Range[250]], # == 2 || # == 11 || MatchQ[Mod[#, 44], Alternatives[7, 19, 35, 39, 43]]&] (* Jean-François Alcover, Oct 28 2016 *)

A141187 Primes of the form -x^2+6xy+3y^2 (as well as of the form 8x^2+12xy+3*y^2).

Original entry on oeis.org

3, 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263, 311, 347, 359, 383, 419, 431, 443, 467, 479, 491, 503, 563, 587, 599, 647, 659, 683, 719, 743, 827, 839, 863, 887, 911, 947, 971, 983, 1019, 1031, 1091, 1103, 1151, 1163, 1187, 1223

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jun 12 2008

nonn

Discriminant = 48. Class = 2. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.
Values of the quadratic form are {0,3,8,11} mod 12, so all values with the exception of 3 are also in A068231. - R. J. Mathar, Jul 30 2008
Is this the same sequence (apart from the initial 3) as A068231? [Yes, since the orders of imaginary quadratic fields with discriminant 48 has 1 class per genus (can be verified by the quadclassunit() function in PARI), so the primes represented by a binary quadratic form of this discriminant are determined by a congruence condition. - Jianing Song, Jun 22 2025]

			a(3)=23 because we can write 23= -1^2+6*1*2+3*2^2 (or 23=8*1^2+12*1*1+3*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A038872 (d=5), A038873 (d=8), A068228 (d=12, 48, or -36), A038883 (d=13), A038889 (d=17), A141111 and A141112 (d=65).
Essentially the same as A068231 and A141123.
Cf. A243169.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -x^2 + 6*x*y + 3*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)

More terms from Colin Barker, Apr 05 2015

A141163 Primes of the form x^2+12xy-y^2.

Original entry on oeis.org

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

Offset: 1

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

nonn

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.

			a(4)=137 because we can write 137= 3^2+12*3*4-4^2.

Z. I. Borevich and I. R. Shafarevich, Number Theory.

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
Peter Luschny, Binary Quadratic Forms
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141161 (d=148).

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

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 *)

# 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

A141167 Primes of the form 8x^2+xy-8*y^2.

Original entry on oeis.org

61, 67, 113, 157, 193, 197, 227, 241, 257, 419, 499, 587, 631, 643, 653, 739, 821, 823, 859, 863, 907, 929, 947, 971, 997, 1019, 1039, 1051, 1087, 1181, 1187, 1217, 1289, 1303, 1319, 1373, 1511, 1531, 1637, 1777, 1783, 1801, 1913, 1997, 2027, 2039, 2069, 2087, 2129, 2213

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jun 12 2008

nonn

Discriminant = 257. 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.

			a(6)=197 because we can write 197 = 8*5^2+5*1-8*1^2.

Z. I. Borevich and I. R. Shafarevich, Number Theory

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
Peter Luschny, Binary Quadratic Forms
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Numbers of the form 8x^2+xy-8y^2 in A243180.
Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141168 (d=257).
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

q := 8*x^2 + x*y - 8*y^2; pmax = 3000; xmax = xmax0 = 50; ymin = ymin0 = -50; ymax = ymax0 = 50; k = 1.3 (* expansion coeff. for maxima *) ; dx = dy = 2; 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, If[xmax == xmax0, xmax, Floor[k*xmax]], dx}, {y, If[ymin == ymin0, ymin, Floor[k*ymin]], If[ymax == ymax0, ymax, Floor[k*ymax]]}, dy]][[2, 1]] // Union; xmax = Max[xx]; ymin = Min[yy]; ymax = Max[yy]; Print[Length[prms], " terms", "  xmax = ", xmax, "  ymin = ", ymin, "  ymax = ", ymax ]]; A141167 = prms (* Jean-François Alcover, Oct 26 2016 *)

# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([8, 1, -8])
print(Q.represented_positives(2213, 'prime')) # Peter Luschny, Oct 26 2016

A141168 Primes of the form 4x^2+9xy-11y^2.

Original entry on oeis.org

2, 11, 13, 17, 23, 29, 31, 59, 73, 79, 89, 137, 139, 173, 199, 211, 223, 239, 283, 293, 307, 317, 349, 373, 379, 397, 401, 433, 457, 479, 491, 503, 523, 563, 571, 593, 613, 647, 673, 683, 701, 709, 719, 727, 769, 773, 787, 797, 829, 839, 887, 911, 967

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jun 12 2008

nonn

Discriminant = 257. 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

Also primes represented by the improperly equivalent form 11*x^2+9*x*y-4*y^2. - Juan Arias-de-Reyna, Mar 18 2011

			a(5)=23 because we can write 23= 4*2^2+9*2*1-11*1^2

Z. I. Borevich and I. R. Shafarevich, Number Theory

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141167 (d=257).

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

A141170 Primes of the form x^2+4xy-2y^2 (as well as of the form 3x^2+6xy+y^2).

Original entry on oeis.org

3, 19, 43, 67, 73, 97, 139, 163, 193, 211, 241, 283, 307, 313, 331, 337, 379, 409, 433, 457, 499, 523, 547, 571, 577, 601, 619, 643, 673, 691, 739, 769, 787, 811, 859, 883, 907, 937, 1009, 1033, 1051, 1123, 1129, 1153, 1171, 1201, 1249, 1291, 1297, 1321, 1459, 1483, 1489, 1531

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 12 2008

nonn

Discriminant = 24. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.
Also, primes of form u^2 - 6v^2. The transformation {u,v} = {x+2y,y} yields the form in the title. - Tito Piezas III, Dec 31 2008
Conjecture: this is also the list of primes that are simultaneously of the form x^2+2y^2 and of the form x^2+3y^2; that is, the intersection of A002476 and A033203. - Zak Seidov, Jun 07 2014
This is also the list of primes p such that p = 3 or p is congruent to 1 or 19 mod 24. - Jean-François Alcover, Oct 28 2016

			a(2)=19 because we can write 19=3^2+4*3*1-2*1^2 (or 19=3*1^2+6*1*2+2^2)

Z. I. Borevich and I. R. Shafarevich, Number Theory.

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A141171 (d=24), A106950 (Primes of the form x^2+18y^2), A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
Cf. also A242661, A002476, A033203.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

xy[{x_, y_}]:={x^2 + 4 x y - 2 y^2, y^2 + 4 x y - 2 x^2}; Union[Select[Flatten[xy/@Subsets[Range[40], {2}]], #>0&&PrimeQ[#]&]] (* Vincenzo Librandi, Jun 09 2014 *)
Select[Prime[Range[250]], # == 3 || MatchQ[Mod[#, 24], 1|19]&] (* Jean-François Alcover, Oct 28 2016 *)

A141172 Primes of the form 2x^2+2x*y-3y^2 (as well as of the form 2x^2+6xy+y^2).

Original entry on oeis.org

2, 29, 37, 53, 109, 113, 137, 149, 193, 197, 233, 277, 281, 317, 337, 373, 389, 401, 421, 449, 457, 541, 557, 569, 613, 617, 641, 653, 673, 701, 709, 757, 809, 821, 877, 953, 977, 1009, 1033, 1061, 1093, 1117, 1129, 1201, 1213, 1229, 1289, 1297, 1373, 1381, 1409, 1429, 1453, 1481, 1493

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (marcanmar(AT)alum.us.es), Jun 12 2008

nonn

Discriminant = 28. Class = 2. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.
Also, primes of form u^2-7v^2. The transformation {u,v}={3x+y,x} yields the second quadratic form given in the title. - Tito Piezas III, Dec 28 2008
This is also the list of primes p such that p = 2 or p is congruent to 1, 9 or 25 mod 28 - Jean-François Alcover, Oct 28 2016

			a(2)=29 because we can write 29=2*4^2+2*4*3-3*3^2 (or 29=2*1^2+6*1*3+3^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A141173 (d=28) A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65).
Cf. also A242662.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Select[Prime[Range[250]], # == 2 || MatchQ[Mod[#, 28], 1|9|25]&] (* Jean-François Alcover, Oct 28 2016 *)

cp's OEIS Frontend

A141165 Primes of the form 9*x^2+7*x*y-5*y^2.

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PARI

Sage

A141166 Primes of the form x^2+15*x*y-y^2.

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A141171 Primes of the form -x^2+4*x*y+2*y^2 (as well as of the form 5*x^2+8*x*y+2*y^2).

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Maple

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A141183 Primes of the form -x^2+6*x*y+2*y^2 (as well as of the form 7*x^2+10*x*y+2*y^2).

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A141187 Primes of the form -x^2+6*x*y+3*y^2 (as well as of the form 8*x^2+12*x*y+3*y^2).

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Extensions

A141163 Primes of the form x^2+12*x*y-y^2.

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Sage

A141167 Primes of the form 8*x^2+x*y-8*y^2.

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A141165 Primes of the form 9x^2+7xy-5y^2.

A141166 Primes of the form x^2+15xy-y^2.

A141171 Primes of the form -x^2+4xy+2y^2 (as well as of the form 5x^2+8xy+2*y^2).

A141183 Primes of the form -x^2+6xy+2y^2 (as well as of the form 7x^2+10xy+2*y^2).

A141187 Primes of the form -x^2+6xy+3y^2 (as well as of the form 8x^2+12xy+3*y^2).

A141163 Primes of the form x^2+12xy-y^2.

A141167 Primes of the form 8x^2+xy-8*y^2.

A141168 Primes of the form 4x^2+9xy-11y^2.

A141170 Primes of the form x^2+4xy-2y^2 (as well as of the form 3x^2+6xy+y^2).

A141172 Primes of the form 2x^2+2x*y-3y^2 (as well as of the form 2x^2+6xy+y^2).