A107132 -id:A107132 - 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

A107145 Primes of the form x^2 + 40y^2.

Original entry on oeis.org

41, 89, 241, 281, 401, 409, 449, 521, 569, 601, 641, 761, 769, 809, 881, 929, 1009, 1049, 1129, 1201, 1249, 1289, 1321, 1361, 1409, 1481, 1489, 1601, 1609, 1721, 1801, 1889, 2081, 2089, 2129, 2161, 2281, 2441, 2521, 2609, 2689, 2729, 2801

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -160. See A107132 for more information.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139643.

[ p: p in PrimesUpTo(3000) | p mod 40 in {1, 9} ]; // Vincenzo Librandi, Jul 24 2012

QuadPrimes2[1, 0, 40, 10000] (* see A106856 *)

list(lim)=my(v=List(),t); forprime(p=41,lim, t=p%40; if(t==1||t==9, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 09 2017

The primes are congruent to {1, 9} (mod 40). - T. D. Noe, Apr 29 2008

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.

A107181 Primes of the form 8x^2 + 9y^2.

Original entry on oeis.org

17, 41, 89, 113, 137, 233, 257, 281, 353, 401, 449, 521, 569, 593, 617, 641, 761, 809, 857, 881, 929, 953, 977, 1049, 1097, 1193, 1217, 1289, 1361, 1409, 1433, 1481, 1553, 1601, 1697, 1721, 1889, 1913, 2081, 2129, 2153, 2273, 2297, 2393, 2417

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -288. See A107132 for more information.
Also primes of the form 9x^2 + 6xy + 17y^2. See A140633. - T. D. Noe, May 19 2008
All terms are of the form x^2 + y^2, see A002144. - Zak Seidov, Jan 26 2014

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Subsequence of A002144 (Pythagorean primes).

Cf. A139827.

[ p: p in PrimesUpTo(5000) | p mod 24 eq 17 ]; // Vincenzo Librandi, Apr 19 2011

QuadPrimes2[8, 0, 9, 10000] (* see A106856 *)

list(lim)=my(v=List()); forprime(p=17,lim, if(p%24==17, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 10 2017

The primes are congruent to 17 (mod 24). - T. D. Noe, May 02 2008

A107169 Primes of the form 3x^2 + 20y^2.

Original entry on oeis.org

3, 23, 47, 83, 107, 167, 227, 263, 347, 383, 443, 467, 503, 563, 587, 647, 683, 743, 827, 863, 887, 947, 983, 1103, 1163, 1187, 1223, 1283, 1307, 1367, 1427, 1487, 1523, 1583, 1607, 1667, 1787, 1823, 1847, 1907, 2003, 2027, 2063, 2087, 2207

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -240. See A107132 for more information.

Except for 3, also primes of the forms 2x^2 + 2xy + 23y^2 (A139831) and 8x^2 + 4xy + 23y^2. See A140633. - T. D. Noe, May 19 2008

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139827.

[3] cat [p: p in PrimesUpTo(3000) | p mod 60 in [23, 47]]; // Vincenzo Librandi, Jul 25 2012

QuadPrimes2[3, 0, 20, 10000] (* see A106856 *)

list(lim)=my(v=List([3]),t); forprime(p=23,lim, t=p%60; if(t==23||t==47, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 10 2017

Except for 3, the primes are congruent to {23, 47} (mod 60). - T. D. Noe, May 02 2008

A107135 Primes of the form 5x^2 + 6y^2.

Original entry on oeis.org

5, 11, 29, 59, 101, 131, 149, 179, 251, 269, 389, 419, 461, 491, 509, 659, 701, 821, 941, 971, 1019, 1061, 1091, 1109, 1181, 1229, 1259, 1301, 1451, 1499, 1571, 1619, 1709, 1811, 1901, 1931, 1949, 1979, 2069, 2099, 2141, 2309, 2339, 2381, 2411

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -120. See A107132 for more information.

Except for 5, also primes of the form 11x^2 + 4xy + 14y^2. See A140633. - T. D. Noe, May 19 2008

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139827.

[ p: p in PrimesUpTo(3000) | p mod 120 in {5, 11, 29, 59, 101} ]; // Vincenzo Librandi, Jul 23 2012

QuadPrimes2[5, 0, 6, 10000] (* see A106856 *)

list(lim)=my(v=List([5]),s=[11,29,59,101]); forprime(p=11,lim, if(setsearch(s,p%120), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 09 2017

The primes are congruent to {5, 11, 29, 59, 101} (mod 120). - T. D. Noe, May 02 2008

A107141 Primes of the form 4x^2 + 9y^2.

Original entry on oeis.org

13, 73, 97, 109, 181, 229, 241, 277, 337, 409, 421, 457, 541, 709, 733, 757, 829, 1009, 1033, 1093, 1117, 1129, 1153, 1213, 1237, 1249, 1381, 1453, 1489, 1597, 1609, 1621, 1669, 1753, 1777, 1873, 2017, 2029, 2089, 2113, 2161, 2221, 2281

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -144. See A107132 for more information.

These appear to be the same as Glaisher's 1889 list of primes == 1 mod 12 that have "negative character". - N. J. A. Sloane, Jul 30 2015

J. W. L. Glaisher, On the square of Euler's series, Proc. London Math. Soc., 21 (1889), 182-194.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

QuadPrimes2[4, 0, 9, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=1, sqrtint(lim\4), w=4*x^2; for(y=1, sqrtint((lim-w)\9), if(isprime(t=w+9*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

A107144 Primes of the form 5x^2 + 8y^2.

Original entry on oeis.org

5, 13, 37, 53, 157, 173, 197, 277, 293, 317, 373, 397, 557, 613, 653, 677, 733, 757, 773, 797, 853, 877, 997, 1013, 1093, 1117, 1213, 1237, 1277, 1373, 1453, 1493, 1597, 1613, 1637, 1693, 1733, 1877, 1933, 1973, 1997, 2053, 2213, 2237, 2293

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -160. See A107132 for more information.

Except for 5, also primes of the form 13x^2 + 8xy + 32y^2. See A140633. - T. D. Noe, May 19 2008

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139827.

[5] cat [ p: p in PrimesUpTo(3000) | p mod 40 in {13, 37} ]; // Vincenzo Librandi, Jul 24 2012

QuadPrimes2[5, 0, 8, 10000] (* see A106856 *)

list(lim)=my(v=List([5]),t); forprime(p=13,lim, t=p%40; if(t==13||t==37, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 09 2017

Except for 5, the primes are congruent to {13, 37} (mod 40). - T. D. Noe, May 02 2008

A107151 Primes of the form 5x^2 + 9y^2.

Original entry on oeis.org

5, 29, 41, 89, 101, 149, 269, 281, 389, 401, 449, 461, 509, 521, 569, 641, 701, 761, 809, 821, 881, 929, 941, 1049, 1061, 1109, 1181, 1229, 1289, 1301, 1361, 1409, 1481, 1601, 1709, 1721, 1889, 1901, 1949, 2069, 2081, 2129, 2141, 2309, 2381

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -180. See A107132 for more information.

Except for 5, also primes of the form 9x^2 + 6xy + 26y^2. See A140633. - T. D. Noe, May 19 2008

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139827.

[5] cat [ p: p in PrimesUpTo(3000) | p mod 60 in {29, 41 } ]; // Vincenzo Librandi, Jul 24 2012

QuadPrimes2[5, 0, 9, 10000] (* see A106856 *)

list(lim)=my(v=List([5]),t); forprime(p=29,lim, t=p%60; if(t==29||t==41, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 09 2017

Except for 5, the primes are congruent to {29, 41} (mod 60). - T. D. Noe, May 02 2008

A107159 Primes of the form 4x^2 + 13y^2.

Original entry on oeis.org

13, 17, 29, 113, 157, 181, 269, 313, 337, 373, 389, 521, 601, 641, 653, 673, 701, 797, 809, 1069, 1109, 1117, 1153, 1213, 1249, 1453, 1481, 1609, 1613, 1621, 1637, 1777, 1933, 1949, 1973, 2053, 2081, 2089, 2129, 2213, 2237, 2297, 2341, 2357, 2393

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -208. See A107132 for more information.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

QuadPrimes2[4, 0, 13, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=0, sqrtint(lim\4), w=4*x^2; for(y=1, sqrtint((lim-w)\13), if(isprime(t=w+13*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

cp's OEIS Frontend

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

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A107145 Primes of the form x^2 + 40y^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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A107181 Primes of the form 8x^2 + 9y^2.

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A107169 Primes of the form 3x^2 + 20y^2.

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A107135 Primes of the form 5x^2 + 6y^2.

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A107141 Primes of the form 4x^2 + 9y^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).