A038883 -id:A038883 - cp's OEIS frontend

A373751 Array read by ascending antidiagonals: p is a term of row A(n) if and only if p is a prime and p is a quadratic residue modulo prime(n).

2, 3, 3, 5, 7, 5, 2, 11, 13, 7, 3, 7, 19, 19, 11, 3, 5, 11, 29, 31, 13, 2, 13, 11, 23, 31, 37, 17, 5, 13, 17, 23, 29, 41, 43, 19, 2, 7, 17, 23, 31, 37, 59, 61, 23, 5, 3, 11, 19, 29, 37, 43, 61, 67, 29, 2, 7, 13, 17, 43, 43, 47, 53, 71, 73, 31, 3, 5, 13, 23, 19, 47, 53, 53, 67, 79, 79, 37

Offset: 1

Peter Luschny, Jun 28 2024

p is a term of A(n) <=> p is prime and there exists an integer q such that q^2 is congruent to p modulo prime(n).

			Note that the cross-references are hints, not assertions about identity.
.
[ n] [ p]
[ 1] [ 2] [ 2,  3,  5,  7, 11, 13, 17, 19, 23, 29, ...  A000040
[ 2] [ 3] [ 3,  7, 13, 19, 31, 37, 43, 61, 67, 73, ...  A007645
[ 3] [ 5] [ 5, 11, 19, 29, 31, 41, 59, 61, 71, 79, ...  A038872
[ 4] [ 7] [ 2,  7, 11, 23, 29, 37, 43, 53, 67, 71, ...  A045373
[ 5] [11] [ 3,  5, 11, 23, 31, 37, 47, 53, 59, 67, ...  A056874
[ 6] [13] [ 3, 13, 17, 23, 29, 43, 53, 61, 79, 101, ..  A038883
[ 7] [17] [ 2, 13, 17, 19, 43, 47, 53, 59, 67, 83, ...  A038889
[ 8] [19] [ 5,  7, 11, 17, 19, 23, 43, 47, 61, 73, ...  A106863
[ 9] [23] [ 2,  3, 13, 23, 29, 31, 41, 47, 59, 71, ...  A296932
[10] [29] [ 5,  7, 13, 23, 29, 53, 59, 67, 71, 83, ...  A038901
[11] [31] [ 2,  5,  7, 19, 31, 41, 47, 59, 67, 71, ...  A267481
[12] [37] [ 3,  7, 11, 37, 41, 47, 53, 67, 71, 73, ...  A038913
[13] [41] [ 2,  5, 23, 31, 37, 41, 43, 59, 61, 73, ...  A038919
[14] [43] [11, 13, 17, 23, 31, 41, 43, 47, 53, 59, ...  A106891
[15] [47] [ 2,  3,  7, 17, 37, 47, 53, 59, 61, 71, ...  A267601
[16] [53] [ 7, 11, 13, 17, 29, 37, 43, 47, 53, 59, ...  A038901
[17] [59] [ 3,  5,  7, 17, 19, 29, 41, 53, 59, 71, ...  A374156
[18] [61] [ 3,  5, 13, 19, 41, 47, 61, 73, 83, 97, ...  A038941
[19] [67] [17, 19, 23, 29, 37, 47, 59, 67, 71, 73, ...  A106933
[20] [71] [ 2,  3,  5, 19, 29, 37, 43, 71, 73, 79, ...
[21] [73] [ 2,  3, 19, 23, 37, 41, 61, 67, 71, 73, ...  A038957
[22] [79] [ 2,  5, 11, 13, 19, 23, 31, 67, 73, 79, ...
[23] [83] [ 3,  7, 11, 17, 23, 29, 31, 37, 41, 59, ...
[24] [89] [ 2,  5, 11, 17, 47, 53, 67, 71, 73, 79, ...  A038977
[25] [97] [ 2,  3, 11, 31, 43, 47, 53, 61, 73, 79, ...  A038987
.
Prime(n) is a term of row n because for all n >= 1, n is a quadratic residue mod n.

Robert G. Wilson v, Table of n, a(n) for n = 1..10011 (the first 141 antidiagonals, flattened).

Family: A217831 (Euclid's triangle), A372726 (Legendre's triangle), A372877 (Jacobi's triangle), A372728 (Kronecker's triangle), A373223 (Gauss' triangle), A373748 (quadratic residue/nonresidue modulo n).
Cf. A374155 (column 1), A373748.
Cf. A000040, A007645, A038872, A045373, A056874, A038883, A038889, A106863, A296932, A038901, A267481, A038913, A038919, A106891, A267601, A038901, A374156, A038941, A106933, A038957, A038977, A038987.

A := proc(n, len) local c, L, a; a := 2; c := 0; L := NULL; while c < len do if NumberTheory:-QuadraticResidue(a, n) = 1 and isprime(a) then L := L,a; c := c + 1 fi; a := a + 1 od; [L] end: seq(print(A(ithprime(n), 10)), n = 1..25);

f[m_, n_] := Block[{p = Prime@ m}, Union[ Join[{p}, Select[ Prime@ Range@ 22, JacobiSymbol[#, If[m > 1, p, 1]] == 1 &]]]][[n]]; Table[f[n, m -n +1], {m, 12}, {n, m, 1, -1}]
(* To read the array by descending antidiagonals, just exchange the first argument with the second in the function "f" called by the "Table"; i.e., Table[ f[m -n +1, n], {m, 12}, {n, m, 1, -1}] *)

A373751_row(n, LIM=99)={ my(q=prime(n)); [p | p <- primes([1,LIM]), issquare( Mod(p, q))] } \\ M. F. Hasler, Jun 29 2024

# The function 'is_quadratic_residue' is defined in A373748.
def A373751_row(n, len):
    return [a for a in range(len) if is_quadratic_residue(a, n) and is_prime(a)]
for p in prime_range(99): print([p], A373751_row(p, 100))

A141178 Primes of the form 3x^2+xy-3y^2 (as well as of the form 3x^2+7xy+y^2).

Original entry on oeis.org

3, 7, 11, 37, 41, 47, 53, 67, 71, 73, 83, 101, 107, 127, 137, 139, 149, 151, 157, 173, 181, 197, 211, 223, 229, 233, 263, 269, 271, 293, 307, 317, 337, 349, 359, 367, 373, 379, 397, 419, 433, 443, 491, 509, 521, 571, 593, 599, 601, 613, 617, 619, 641, 659, 673

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 = 37. Class = 1. 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(3) = 11 because we can write 11 = 3*2^2+2*1-3*1^2 (or 11 = 3*1^2+7*1*1+1^2).

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

Jean-François Alcover, 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)
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer-Verlag Berlin Heidelberg, 1981.

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65).
Primes in A035267.
A subsequence of (and may possibly coincide with) A038913. - R. J. Mathar, Jul 22 2008
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 < 1000, p = NextPrime[p], If[FindInstance[p == 3*x^2 + x*y - 3*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]]
(* or: *)
Select[Prime[Range[200]], # == 37 || MatchQ[Mod[#, 37], Alternatives[1, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25, 26, 27, 28, 30, 33, 34, 36]]&](* Jean-François Alcover, Oct 25 2016, updated Oct 30 2016 *)

A141191 Primes of the form -2x^2+4xy+5y^2 (as well as of the form 10x^2+16xy+5y^2).

Original entry on oeis.org

5, 7, 13, 31, 47, 61, 101, 103, 157, 167, 173, 181, 199, 223, 229, 269, 271, 293, 311, 349, 367, 383, 397, 439, 461, 479, 503, 509, 607, 647, 661, 677, 719, 727, 733, 773, 797, 829, 839, 853, 887, 941, 983, 997, 1013, 1021, 1039, 1063, 1069, 1109, 1151, 1181

Offset: 1

Laura Caballero Fernandez,Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (lourdescm84(AT)hotmail.com), Jun 12 2008

nonn

Discriminant = 56. 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 the form -x^2+6xy+5y^2. cf. A243187.

			a(4)=31 because we can write 31=-2*7^2+4*7*3+5*3^2 (or 31=10*1^2+16*1*1+5*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. A141190 (d=56) A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
Cf. A243187, A243186, A141190.
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 == -2*x^2 + 4*x*y + 5*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

A141215 Primes of the form 3x^2+5x*y-3y^2 (as well as 5x^2+9xy+y^2).

Original entry on oeis.org

3, 5, 13, 19, 41, 47, 61, 73, 83, 97, 103, 107, 109, 113, 127, 131, 137, 149, 163, 167, 179, 197, 199, 229, 239, 241, 257, 263, 269, 271, 283, 293, 317, 347, 353, 367, 379, 431, 439, 443, 449, 461, 463, 479, 487, 491, 503, 563, 569, 571, 601, 607, 613, 619

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 14 2008

nonn

Discriminant = 61. Class = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.
A subsequence of (and may possibly coincide with) A038941. - R. J. Mathar, Jul 22 2008
3*x^2+5*x*y-3*y^2 and 5*x^2+9*x*y+y^2 are equivalent forms.
Also, primes of the form x^2 - 61y^2, of discriminant 244.

			a(8) = 73 because we can write 73 = 3*4^2+5*4*5-3*5^2 (or 73 = 5*3^2+9*3*1+1^2).

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

Robert Israel, 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).
Primes in A243654.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

select(p -> isprime(p) and nops([isolve(x^2 - 61*y^2 = p)])>0, [seq(2*i+1,i=1..1000)]); # Robert Israel, Jun 11 2014

terms = 100; d = 61;
Table[3*x^2 + 5*x*y - 3*y^2, {x, 1, terms}, {y, Floor[(5 - Sqrt[d])*x/6], Ceiling[(5 + Sqrt[d])*x/6]}] // Flatten // Select[#, Positive[#] && PrimeQ[#]&]& // Union // Take[#, terms]& (* Jean-François Alcover, Feb 28 2019 *)

A141160 Primes of the form -x^2 + 3xy + 3y^2 (as well as of the form 5x^2 + 9xy + 3*y^2).

Original entry on oeis.org

3, 5, 17, 41, 47, 59, 83, 89, 101, 131, 167, 173, 227, 251, 257, 269, 293, 311, 353, 383, 419, 461, 467, 479, 503, 509, 521, 563, 587, 593, 647, 677, 719, 761, 773, 797, 839, 857, 881, 887, 929, 941, 971, 983, 1013, 1049, 1091, 1097, 1109, 1151, 1181, 1193

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 = 21. Class number = 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 (primitive).
Except a(1) = 3, primes congruent to {5, 17, 20} mod 21. - Vincenzo Librandi, Jul 11 2018
The comment above is true since the binary quadratic forms with discriminant 21 are in two classes as well as two genera, so there is one class in each genus. A141159 is in the other genus, with primes = 7 or congruent to {1, 4, 16} mod 21. - Jianing Song, Jul 12 2018
4*a(n) can be written in the form 21*w^2 - z^2. - Bruno Berselli, Jul 13 2018
Both forms [-1, 3, 3] (reduced) and [5, 9, 3] (not reduced) are properly (via a determinant +1 matrix) equivalent to the reduced form [3, 3, -1], a member of the 2-cycle [[3, 3, -1], [-1, 3, 3]]. The other reduced form is the principal form [1, 3, -3], with 2-cycle [[1, 3, -3], [-3, 3, 1]] (see, e.g., A141159, A139492). - Wolfdieter Lang, Jun 24 2019

			a(3)=17 because we can write 17 = -1^2 + 3*1*2 + 3*2^2 (or 17 = 5*1^2 + 9*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.

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

Cf. A141159, A139492 (d=21) A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17): A141111, A141112 (d=65).
Primes in A237351.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

[3] cat [p: p in PrimesUpTo(2000) | p mod 21 in [5, 17, 20]]; // Vincenzo Librandi, Jul 11 2018

Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -x^2 + 3*x*y + 3*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)
Join[{3}, Select[Prime[Range[250]], MemberQ[{5, 17, 20}, Mod[#, 21]] &]] (* Vincenzo Librandi, Jul 11 2018 *)

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([-1, 3, 3])
Q.represented_positives(1200, 'prime') # Peter Luschny, Jun 24 2019

More terms from Colin Barker, Apr 05 2015

A141181 Primes of the form 2x^2+3x*y-4y^2 (as well as of the form 2x^2+7xy+y^2).

Original entry on oeis.org

2, 5, 23, 31, 37, 41, 43, 59, 61, 73, 83, 103, 107, 113, 127, 131, 139, 163, 173, 197, 223, 241, 251, 269, 271, 277, 283, 307, 337, 349, 353, 359, 367, 373, 379, 389, 401, 409, 419, 431, 433, 443, 449, 461, 467, 487, 491, 523, 541, 569, 599, 607, 613, 617, 619

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 = 41. Class = 1. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.

It appears that this is the same as "Primes that are squares (mod 41)", cf. A038919 and A373751. - M. F. Hasler, Jun 29 2024

			a(3) = 23 because we can write 23 = 2*3^2+3*3*1-4*1^2 (or 23 = 2*2^2+7*2*1+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, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
Primes in A035269.
A subsequence of (and may possibly coincide with) A038919. - R. J. Mathar, Jul 22 2008
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 < 1000, p = NextPrime[p], If[FindInstance[p == 2*x^2 + 3*x*y - 4*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)

select(p->isprime(p)&&qfbsolve(Qfb(1,7,2),p),[1..1500]) \\ This is to provide a generic characteristic function ("is_A141181") as 1st arg of select(), there are other ways to produce the sequence more efficiently. - M. F. Hasler, Jan 15 2016

A141189 Primes of the form x^2+7xy-y^2 (as well as of the form 7x^2+9x*y+y^2).

Original entry on oeis.org

7, 11, 13, 17, 29, 37, 43, 47, 53, 59, 89, 97, 107, 113, 131, 149, 163, 197, 199, 211, 223, 227, 229, 241, 269, 271, 281, 293, 307, 311, 317, 331, 347, 367, 409, 431, 433, 439, 449, 461, 467, 487, 521, 523, 541, 547, 577, 587, 593, 599, 607, 619, 643, 647, 653

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 = 53. Class = 1. 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 subsequence of (and may possibly coincide with) A038931. - R. J. Mathar, Jul 22 2008

			a(5) = 29 because we can write 29 = 3^2+7*3*1-1^2 (or 29 = 7*1^2+9*1*2+2^2).

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). A038931, A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65).
Primes in A243191.
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 < 1000, p = NextPrime[p], If[FindInstance[p == x^2 + 7*x*y - y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)

A296937 Rational primes that decompose in the field Q(sqrt(13)).

Original entry on oeis.org

3, 17, 23, 29, 43, 53, 61, 79, 101, 103, 107, 113, 127, 131, 139, 157, 173, 179, 181, 191, 199, 211, 233, 251, 257, 263, 269, 277, 283, 311, 313, 337, 347, 367, 373, 389, 419, 433, 439, 443, 467, 491, 503, 521, 523, 547, 563, 569, 571, 599, 601, 607, 641

Offset: 1

N. J. A. Sloane, Dec 26 2017

Is this the same sequence as A141188 or A038883? - R. J. Mathar, Jan 02 2018
From Jianing Song, Apr 21 2022: (Start)
Primes p such that Kronecker(13, p) = Kronecker(p, 13) = 1, where Kronecker() is the Kronecker symbol. That is to say, primes p that are quadratic residues modulo 13.
Primes p such that p^6 == 1 (mod 13).
Primes p == 1, 3, 4, 9, 10, 12 (mod 13). (End)

Cf. A011583 (kronecker symbol modulo 13), A038883.
Rational primes that decompose in the quadratic field with discriminant D: A139513 (D=-20), A191019 (D=-19), A191018 (D=-15), A296920 (D=-11), A033200 (D=-8), A045386 (D=-7), A002144 (D=-4), A002476 (D=-3), A045468 (D=5), A001132 (D=8), A097933 (D=12), this sequence (D=13), A296938 (D=17).
Cf. A038884 (inert rational primes in the field Q(sqrt(13))).

Load the Maple program HH given in A296920. Then run HH(13, 200); This produces A296937, A038884, A038883.

isA296937(p) = isprime(p) && kronecker(p,13) == 1 \\ Jianing Song, Apr 21 2022

Equals A038883 \ {13}. - Jianing Song, Apr 21 2022

A035256 Positive integers of the form x^2+3xy-y^2.

Original entry on oeis.org

1, 3, 4, 9, 12, 13, 16, 17, 23, 25, 27, 29, 36, 39, 43, 48, 49, 51, 52, 53, 61, 64, 68, 69, 75, 79, 81, 87, 92, 100, 101, 103, 107, 108, 113, 116, 117, 121, 127, 129, 131, 139, 144, 147, 153, 156, 157, 159, 169, 172, 173, 179, 181, 183, 191, 192, 196, 199, 204, 207, 208, 211, 212, 221, 225, 233, 237, 243, 244, 251

Offset: 1

N. J. A. Sloane

nonn

This is an indefinite quadratic form of discriminant 13.
Also, positive integers of the form x^2+6xy-4y^2 (an indefinite quadratic form of discriminant 52).
Also, indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m= 13.
From Klaus Purath, May 07 2023: (Start)
Also, positive integers of the form x^2 + (2m+1)xy + (m^2+m-3)y^2, m, x, y integers. This includes the form in the name.
Also, positive integers of the form x^2 + 2mxy + (m^2-13)y^2, m, x, y integers. This includes the form in the comment above.
This sequence contains all squares. The prime factors of the terms except for {2, 5, 7, 11, 19, ...} = A038884 are terms of the sequence. Also the products of terms belong to the sequence. Thus this set of terms is closed under multiplication.
A positive integer N belongs to the sequence if and only if N (modulo 13) is a term of A010376 and, moreover, in the case that prime factors p of N are terms of A038884, they occur only with even exponents. For these prime factors also p (modulo 13) = {2, 5, 6, 7, 8, 11} applies. (End)

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

Primes in this sequence = A038883 and A141188.

Cf. A035195.

formQ[n_] := Reduce[a > 0 && b > 0 && n == a^2 + 3 a*b - b^2, {a, b}, Integers] =!= False; Select[Range[100], formQ] (* Wesley Ivan Hurt, Jun 18 2014 *)

m=13; select(x -> x, direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020

Entry revised by N. J. A. Sloane, Jun 01 2014

A141750 Primes of the form 4x^2 + 3x*y - 4y^2 (as well as of the form 2x^2 + 9xy + y^2).

Original entry on oeis.org

2, 3, 19, 23, 37, 41, 61, 67, 71, 73, 79, 89, 97, 109, 127, 137, 149, 173, 181, 211, 223, 227, 251, 257, 269, 283, 293, 311, 317, 347, 349, 353, 359, 367, 373, 383, 389, 397, 401, 419, 439, 457, 461, 463, 479, 487, 499, 503, 509, 523, 547, 557, 587, 593, 607

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jul 03 2008

nonn

Discriminant = 73. Class = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2-4ac.

Is this the same as A038957? - R. J. Mathar, Jul 04 2008. Answer: almost certainly - see the Tunnell notes in A033212. - N. J. A. Sloane, Oct 18 2014

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

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

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.

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). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).

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A373751 Array read by ascending antidiagonals: p is a term of row A(n) if and only if p is a prime and p is a quadratic residue modulo prime(n).

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A141178 Primes of the form 3*x^2+x*y-3*y^2 (as well as of the form 3*x^2+7*x*y+y^2).

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A141191 Primes of the form -2*x^2+4*x*y+5*y^2 (as well as of the form 10*x^2+16*x*y+5*y^2).

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A141215 Primes of the form 3*x^2+5*x*y-3*y^2 (as well as 5*x^2+9*x*y+y^2).

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A141160 Primes of the form -x^2 + 3*x*y + 3*y^2 (as well as of the form 5*x^2 + 9*x*y + 3*y^2).

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A141181 Primes of the form 2*x^2+3*x*y-4*y^2 (as well as of the form 2*x^2+7*x*y+y^2).

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A141189 Primes of the form x^2+7*x*y-y^2 (as well as of the form 7*x^2+9*x*y+y^2).

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A141178 Primes of the form 3x^2+xy-3y^2 (as well as of the form 3x^2+7xy+y^2).

A141191 Primes of the form -2x^2+4xy+5y^2 (as well as of the form 10x^2+16xy+5y^2).

A141215 Primes of the form 3x^2+5x*y-3y^2 (as well as 5x^2+9xy+y^2).

A141160 Primes of the form -x^2 + 3xy + 3y^2 (as well as of the form 5x^2 + 9xy + 3*y^2).

A141181 Primes of the form 2x^2+3x*y-4y^2 (as well as of the form 2x^2+7xy+y^2).

A141189 Primes of the form x^2+7xy-y^2 (as well as of the form 7x^2+9x*y+y^2).

A141750 Primes of the form 4x^2 + 3x*y - 4y^2 (as well as of the form 2x^2 + 9xy + y^2).