A141303 -id:A141303 - cp's OEIS frontend

A107152 Primes of the form x^2 + 45y^2.

61, 109, 181, 229, 241, 349, 409, 421, 541, 601, 661, 709, 769, 829, 1009, 1021, 1069, 1129, 1201, 1249, 1321, 1381, 1429, 1489, 1549, 1609, 1621, 1669, 1741, 1789, 1801, 1861, 2029, 2089, 2161, 2221, 2269, 2281, 2341, 2389, 2521, 2689, 2749, 3001, 3049, 3061, 3109, 3121, 3169, 3181

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -180. See A107132 for more information.
Also primes of the form x^2 + 60y^2. See A140633. - T. D. Noe, May 19 2008
Also primes of the form x^2+6*x*y-6*y^2, of discriminant 60 (as well as of the form x^2+8*x*y+y^2). -  Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 24 2008

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]
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. A139643.
Cf. A141302, A141303, A141304 (d=60).
All representatives in A243188.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

[ p: p in PrimesUpTo(3000) | p mod 60 in {1, 49 } ]; // Vincenzo Librandi, Jul 24 2012

QuadPrimes2[1, 0, 45, 10000] (* see A106856 *)
Select[Prime[Range[500]], MatchQ[Mod[#, 60], 1|49]&] (* Jean-François Alcover, Oct 28 2016 *)

list(lim)=my(v=List(),t); forprime(p=61,lim, t=p%60; if(t==1||t==49, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 09 2017

Primes congruent to {1, 49} (mod 60). - T. D. Noe, Apr 29 2008

A141302 Primes of the form -x^2+6xy+6y^2 (as well as of the form 11x^2+18xy+6*y^2).

Original entry on oeis.org

11, 59, 71, 131, 179, 191, 239, 251, 311, 359, 419, 431, 479, 491, 599, 659, 719, 839, 911, 971, 1019, 1031, 1091, 1151, 1259, 1319, 1439, 1451, 1499, 1511, 1559, 1571, 1619, 1811, 1871, 1931, 1979, 2039, 2099, 2111, 2339, 2351, 2399, 2411, 2459, 2531, 2579, 2591, 2699, 2711

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 24 2008

nonn

Discriminant = 60. Class number = 4. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1 if they are primitive.

The Pell form X^2 - 15*Y^2 represents the negative primes -a(n), for n >= 1. - Wolfdieter Lang, Nov 28 2024

			a(3)=71 because we can write 71=-1^2+6*1*3+6*3^2 (or 71=11*1^2+18*1*2+6*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. A107152, A141303, A141304 (d=60).

Primes in A237606.

Reap[For[p = 2, p < 3000, p = NextPrime[p], If[FindInstance[p == -x^2 + 6*x*y + 6*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2016 *)

Primes congruent to {11, 59} (mod 60). -Wolfdieter Lang, Dec 22 2024

Offset corrected by Mohammed Yaseen, May 20 2023

A141304 Primes of the form -2x^2+6xy+3y^2 (as well as of the form 7x^2+12xy+3y^2).

Original entry on oeis.org

3, 7, 43, 67, 103, 127, 163, 223, 283, 307, 367, 463, 487, 523, 547, 607, 643, 727, 787, 823, 883, 907, 967, 1063, 1087, 1123, 1303, 1327, 1423, 1447, 1483, 1543, 1567, 1627, 1663, 1723, 1747, 1783, 1867, 1987, 2083, 2143, 2203, 2287, 2347, 2383, 2467, 2503, 2647, 2683

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 24 2008

nonn

Discriminant = 60. Class = 4. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1

This is also the list of primes p such that p = 3 or p is congruent to 7 or 43 mod 60. - Jean-François Alcover, Oct 28 2016

			a(3)=43 because we can write 43=-2*1^2+6*1*3+3*3^2 (or 43=7*1^2+12*1*2+3*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. A107152, A141302, A141303 (d=60).

Primes in A243190.

Select[Prime[Range[500]], # == 3 || MatchQ[Mod[#, 60], 7|43]&] (* Jean-François Alcover, Oct 28 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.

A243189 Nonnegative numbers of the form 2x^2 + 6xy - 3y^2.

Original entry on oeis.org

0, 2, 5, 8, 17, 18, 20, 32, 33, 42, 45, 50, 53, 68, 72, 77, 80, 98, 105, 113, 122, 125, 128, 132, 137, 153, 162, 168, 170, 173, 177, 180, 197, 200, 212, 213, 218, 233, 242, 245, 257, 258, 272, 288, 293, 297, 305, 308, 317, 320, 330, 338, 353, 357, 362, 378

Offset: 1

N. J. A. Sloane, Jun 05 2014

nonn

Discriminant 60.
Nonnegative numbers of the form 5x^2 - 3y^2. - Jon E. Schoenfield, Jun 03 2022
From Klaus Purath, Jul 26 2023: (Start)
Nonnegative integers k such that 3x^2 - 5y^2 + k = 0 has integer solutions.
Also nonnegative integers of the form 2x^2 + (4m+2)xy + (2m^2+2m-7)y^2 for integers m. This includes the form in the name with m = 1.
Also nonnegative integers of the form 5x^2 + 10mxy + (5m^2-3)y^2 for integers m. This includes the form from Jon E. Schoenfield above with m = 0.
There are no squares in this sequence. Even powers of terms as well as products of an even number of terms belong to A243188.
Odd powers of terms as well as products of an odd number of terms belong to the sequence. This can be proved with respect to the form 5x^2 - 3y^2 by the following identity: (na^2 - kb^2)(nc^2 - kd^2)(ne^2 - kf^2) = n[a(nce + kdf) + bk(cf + de)]^2 - k[na(cf + de) + b(nce + kdf)]^2 for all a, b, c, d, e, f, k, n in R. This can be verified by expanding both sides of the equation.
(End)

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

Primes: A141303. Cf. A243188, A107152, A237606, A141302, A243190, A141304.

Reap[For[n = 0, n <= 200, n++, If[Reduce[2*x^2 + 6*x*y - 3*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]]

0 prepended and more terms from Colin Barker, Apr 07 2015

A243190 Nonnegative numbers of the form -2x^2+6xy+3y^2.

Original entry on oeis.org

0, 3, 7, 12, 22, 27, 28, 30, 43, 48, 55, 63, 67, 70, 75, 88, 102, 103, 108, 112, 118, 120, 127, 142, 147, 163, 172, 175, 183, 187, 192, 198, 220, 223, 238, 243, 252, 255, 262, 268, 270, 280, 283, 295, 300, 307, 318, 327, 343, 352, 355, 358, 363, 367, 382

Offset: 1

N. J. A. Sloane, Jun 05 2014

nonn

Discriminant 60.

Also: nonnegative 3x^2-5y^2 since 3y^2+6xy-2x^2 = 3(y+x)^2-5x^2. - R. J. Mathar, Jun 10 2020

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

Primes: A141304. Cf. A243188, A107152, A237606, A141302, A243189, A141303.

Reap[For[n = 0, n <= 200, n++, If[Reduce[-2*x^2 + 6*x*y + 3*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]]

0 prepended and more terms from Colin Barker, Apr 07 2015

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

cp's OEIS Frontend

A107152 Primes of the form x^2 + 45y^2.

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A141302 Primes of the form -x^2+6*x*y+6*y^2 (as well as of the form 11*x^2+18*x*y+6*y^2).

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A141304 Primes of the form -2*x^2+6*x*y+3*y^2 (as well as of the form 7*x^2+12*x*y+3*y^2).

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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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A243189 Nonnegative numbers of the form 2x^2 + 6xy - 3y^2.

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A243190 Nonnegative numbers of the form -2x^2+6xy+3y^2.

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Mathematica

Extensions

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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A141302 Primes of the form -x^2+6xy+6y^2 (as well as of the form 11x^2+18xy+6*y^2).

A141304 Primes of the form -2x^2+6xy+3y^2 (as well as of the form 7x^2+12xy+3y^2).

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).

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).