A141163 -id:A141163 - cp's OEIS frontend

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).

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

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).

A141772 Primes of the form 3x^2 + 5xy - 5y^2 (as well as of the form 7x^2 + 13xy + 3y^2).

Original entry on oeis.org

3, 5, 7, 17, 23, 37, 73, 97, 107, 113, 163, 167, 173, 193, 197, 227, 233, 277, 283, 313, 317, 337, 347, 367, 397, 487, 503, 547, 607, 617, 643, 653, 673, 677, 683, 743, 787, 823, 827, 853, 857, 877, 887, 907, 947, 983, 997, 1013, 1093, 1117, 1153, 1163, 1187

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 04 2008

nonn

Discriminant = 85. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.

			a(1) = 3 because we can write 3 = 3*1^2 + 5*1*0 - 5*0^2 (or 3 = 7*0^2 + 13*0*1 + 3*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.

Cf. A141773 (d=85). 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). A141750 (d=73). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).

More terms from Colin Barker, Apr 04 2015

Typo in crossrefs fixed by Colin Barker, Apr 05 2015

A141778 Primes of the form 4x^2 + 3x*y - 5y^2 (as well as of the form 8x^2 + 11xy + y^2).

Original entry on oeis.org

2, 5, 11, 17, 47, 53, 67, 71, 73, 79, 89, 97, 107, 109, 131, 139, 157, 167, 173, 179, 199, 223, 227, 233, 251, 257, 263, 269, 271, 277, 283, 307, 311, 317, 331, 347, 367, 373, 401, 409, 443, 449, 461, 463, 467, 479, 487, 509, 523, 587, 601, 607, 613, 619, 631

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 04 2008

nonn

Discriminant = 89. 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) A038977. - R. J. Mathar, Jul 22 2008

			a(1) = 2 because we can write 2 = 4*1^2 + 3*1*1 - 5*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). A141750 (d=73). A141772, A141773 (d=85). A141776, A141777 (d=88). A141778 (d=89). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).

Typo in crossrefs fixed by Colin Barker, Apr 05 2015

A309134 Nonnegative integers of the form x^2+12xy-y^2.

Original entry on oeis.org

0, 1, 4, 9, 12, 16, 21, 25, 27, 28, 33, 36, 37, 44, 48, 49, 63, 64, 67, 77, 81, 84, 99, 100, 107, 108, 112, 121, 123, 132, 137, 139, 141, 144, 147, 148, 151, 159, 164, 169, 176, 188, 189, 192, 196, 212, 213, 219, 225, 231, 233, 243, 249, 252, 256, 268, 269, 284

Offset: 1

Hugo Pfoertner, Jul 14 2019

nonn

Discriminant of indefinite binary quadratic form: 148.

Will Jagy, C++ program Conway_Positive_All.cc to find all positive numbers represented by an indefinite binary quadratic form
Peter Luschny, Binary Quadratic Forms
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS(Index to related sequences, programs, references)

Primes in this sequence: A141163.

cp's OEIS Frontend

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

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A141772 Primes of the form 3*x^2 + 5*x*y - 5*y^2 (as well as of the form 7*x^2 + 13*x*y + 3*y^2).

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A141778 Primes of the form 4*x^2 + 3*x*y - 5*y^2 (as well as of the form 8*x^2 + 11*x*y + y^2).

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A309134 Nonnegative integers of the form x^2+12*x*y-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.

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

A141772 Primes of the form 3x^2 + 5xy - 5y^2 (as well as of the form 7x^2 + 13xy + 3y^2).

A141778 Primes of the form 4x^2 + 3x*y - 5y^2 (as well as of the form 8x^2 + 11xy + y^2).

A309134 Nonnegative integers of the form x^2+12xy-y^2.