A038889 -id:A038889 - cp's OEIS frontend

A191025 Primes p which have Kronecker symbol (p|34) = 1.

3, 5, 11, 29, 37, 47, 61, 89, 103, 107, 109, 127, 131, 137, 139, 151, 163, 173, 181, 191, 197, 211, 223, 227, 239, 257, 263, 269, 271, 277, 281, 283, 317, 347, 353, 359, 379, 383, 397, 409, 419, 433, 457, 463, 499, 541, 547, 569, 571, 577, 593, 599, 619, 631

Offset: 1

T. D. Noe, May 24 2011

Originally incorrectly named "primes which are squares (mod 34)", which is sequence A038889. - M. F. Hasler, Jan 15 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A191017, A191018, A191020, A191023.

[p: p in PrimesUpTo(631) | KroneckerSymbol(p, 34) eq 1]; // Vincenzo Librandi, Sep 11 2012

Select[Prime[Range[200]], JacobiSymbol[#,34]==1&]

Definition corrected (following an observation by David Broadhurst) by M. F. Hasler, Jan 15 2016

A141373 Primes of the form 3x^2+16y^2. Also primes of the form 4x^2+4xy-5y^2 (as well as primes the form 4x^2+12xy+3y^2).

Original entry on oeis.org

3, 19, 43, 67, 139, 163, 211, 283, 307, 331, 379, 499, 523, 547, 571, 619, 643, 691, 739, 787, 811, 859, 883, 907, 1051, 1123, 1171, 1291, 1459, 1483, 1531, 1579, 1627, 1699, 1723, 1747, 1867, 1987, 2011, 2083, 2131, 2179, 2203, 2251, 2347, 2371, 2467, 2539

Offset: 1

T. D. Noe, May 13 2005; Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 28 2008

nonn

The discriminant is -192 (or 96, or ...), depending on which quadratic form is used for the definition. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1. See A107132 for more information.

Except for 3, also primes of the forms 4x^2 + 4xy + 19y^2 and 16x^2 + 8xy + 19y^2. See A140633. - T. D. Noe, May 19 2008

			19 is a member because we can write 19=4*2^2+4*2*1-5*1^2 (or 19=4*1^2+12*1*1+3*1^2).

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

Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy] See Table II.
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. A107132, A139827, A107003, A141375, A141376 (d=96).
See also A038872 (d=5),
A038873 (d=8),
A068228, A141123 (d=12),
A038883 (d=13),
A038889 (d=17),
A141158 (d=20),
A141159, A141160 (d=21),
A141170, A141171 (d=24),
A141172, A141173 (d=28),
A141174, A141175 (d=32),
A141176, A141177 (d=33),
A141178 (d=37),
A141179, A141180 (d=40),
A141181 (d=41),
A141182, A141183 (d=44),
A033212, A141785 (d=45),
A068228, A141187 (d=48),
A141188 (d=52),
A141189 (d=53),
A141190, A141191 (d=56),
A141192, A141193 (d=57),
A107152, A141302, A141303, A141304 (d=60),
A141215 (d=61),
A141111, A141112 (d=65),
A141336, A141337 (d=92),
A141338, A141339 (d=93),
A141161, A141163 (d=148),
A141165, A141166 (d=229),
A141167, A141167, A141167 (d=257).

[3] cat [ p: p in PrimesUpTo(3000) | p mod 24 in {19 } ]; // Vincenzo Librandi, Jul 24 2012

QuadPrimes2[3, 0, 16, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=1, sqrtint(lim\3), w=3*x^2; for(y=0, sqrtint((lim-w)\16), if(isprime(t=w+16*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

Except for 3, the primes are congruent to 19 (mod 24). - T. D. Noe, May 02 2008

More terms from Colin Barker, Apr 05 2015

Edited by N. J. A. Sloane, Jul 14 2019, combining two identical entries both with multiple cross-references.

A141161 Primes of the form 4x^2+6xy-7y^2.

Original entry on oeis.org

3, 7, 11, 41, 47, 53, 71, 73, 83, 101, 127, 149, 157, 173, 181, 197, 211, 223, 229, 263, 271, 307, 337, 359, 373, 379, 397, 419, 433, 443, 509, 521, 571, 593, 599, 613, 617, 619, 641, 659, 673, 677, 719, 733, 739, 743, 751, 761, 773, 787, 811, 821, 887, 937

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.

Also primes represented by the improperly equivalent form 7*x^2 + 6*x*y - 4*y^2

			a(8)=73 because we can write 73= 4*4^2+6*4*3-7*3^2.

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). A141163 (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 := 4*x^2 + 6*x*y - 7*y^2; pmax = 1000; 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 ]]; A141161 = prms (* Jean-François Alcover, Oct 26 2016 *)

# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([4, 6, -7])
Q.represented_positives(937, 'prime')  # Peter Luschny, Oct 26 2016

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

Original entry on oeis.org

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

cp's OEIS Frontend

A191025 Primes p which have Kronecker symbol (p|34) = 1.

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A141373 Primes of the form 3*x^2+16*y^2. Also primes of the form 4*x^2+4*x*y-5*y^2 (as well as primes the form 4*x^2+12*x*y+3*y^2).

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A141161 Primes of the form 4*x^2+6*x*y-7*y^2.

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

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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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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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A141373 Primes of the form 3x^2+16y^2. Also primes of the form 4x^2+4xy-5y^2 (as well as primes the form 4x^2+12xy+3y^2).

A141161 Primes of the form 4x^2+6xy-7y^2.

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.