A106856 -id:A106856 - cp's OEIS frontend

A106864 Primes of the form 2x^2+2xy+3y^2 with x and y nonnegative.

2, 3, 7, 43, 47, 83, 103, 107, 163, 167, 223, 263, 283, 347, 367, 383, 443, 467, 487, 503, 523, 547, 607, 643, 683, 743, 787, 823, 827, 863, 883, 887, 947, 967, 983, 1063, 1087, 1123, 1223, 1303, 1307, 1367, 1423, 1427, 1447, 1483, 1487, 1523, 1583

Offset: 1

T. D. Noe, May 09 2005

See A106865 for all primes represented by this form.
Discriminant=-20.
Union of {2} and primes == 3 or 7 mod 20. - N. J. A. Sloane, Sep 04 2012

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989. See Eq. (2.22), p. 33. - N. J. A. Sloane, Sep 04 2012

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. A106865.

QuadPrimes2[2, 2, 3, 10000] (* see A106856 *)

A106878 Primes of the form 3x^2+xy+3y^2.

Original entry on oeis.org

3, 5, 7, 13, 17, 47, 73, 83, 97, 103, 157, 167, 173, 223, 227, 257, 283, 293, 307, 313, 353, 367, 383, 397, 433, 467, 503, 523, 563, 577, 587, 593, 607, 643, 647, 677, 727, 733, 773, 787, 797, 853, 857, 887, 937, 983, 997, 1013, 1063, 1097, 1123, 1153

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-35.

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)

Primes in A243179.

Union[QuadPrimes2[3, 1, 3, 10000], QuadPrimes2[3, -1, 3, 10000]] (* see A106856 *)

A106889 Primes of the form 2x^2 + 5y^2.

Original entry on oeis.org

2, 5, 7, 13, 23, 37, 47, 53, 103, 127, 157, 167, 173, 197, 223, 263, 277, 293, 317, 367, 373, 383, 397, 463, 487, 503, 557, 607, 613, 647, 653, 677, 727, 733, 743, 757, 773, 797, 823, 853, 863, 877, 887, 967, 983, 997, 1013, 1063, 1087, 1093, 1103, 1117

Offset: 1

T. D. Noe, May 09 2005

Discriminant = -40.

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. Primes in A020674.

QuadPrimes2[2, 0, 5, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=0, sqrtint(lim\2), w=2*x^2; for(y=0, sqrtint((lim-w)\5), if(isprime(t=w+5*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

The primes are congruent to {2, 5, 7, 13, 23, 37} (mod 40). - T. D. Noe, May 02 2008

A106950 Primes of the form x^2 + 18y^2.

Original entry on oeis.org

19, 43, 67, 73, 97, 139, 163, 193, 211, 241, 283, 307, 313, 331, 337, 379, 409, 433, 457, 499, 523, 547, 571, 577, 601, 619, 643, 673, 691, 739, 769, 787, 811, 859, 883, 907, 937, 1009, 1033, 1051, 1123, 1129, 1153, 1171, 1201, 1249, 1291, 1297, 1321

Offset: 1

T. D. Noe, May 09 2005

Discriminant = -72.

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)

[ p: p in PrimesUpTo(2000) | p mod 24 in {1, 19} ]; // Vincenzo Librandi, Jul 22 2012

QuadPrimes2[1, 0, 18, 10000] (* see A106856 *)

{a(n)= my(m, c); if(n<1, 0, c=0; m=0; while( cMichael Somos, Aug 19 2006 */

Primes congruent to 1,19 modulo 24. - Michael Somos, Aug 19 2006

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

A139855 Primes of the form 4x^2+4xy+31y^2.

Original entry on oeis.org

31, 79, 151, 199, 271, 439, 631, 751, 919, 991, 1039, 1231, 1279, 1399, 1471, 1759, 1831, 1879, 1951, 1999, 2239, 2311, 2551, 2671, 2719, 2791, 3079, 3271, 3319, 3391, 3511, 3559, 3631, 3919, 4111, 4159, 4231, 4519, 4591, 4639, 4759, 4831

Offset: 1

T. D. Noe, May 02 2008

Discriminant = -480. See A139827 for more information.
Also primes of the form 15x^2+16y^2, which has discriminant = -960. - T. D. Noe, May 07 2008
Also primes of the form 16x^2+8xy+31y^2, which has discriminant = -1920. See A140633. - T. D. Noe, May 19 2008

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 item 14 in Table II.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

[ p: p in PrimesUpTo(6000) | p mod 120 in {31, 79}]; // Vincenzo Librandi, Jul 29 2012

QuadPrimes2[4, -4, 31, 10000] (* see A106856 *)

The primes are congruent to {31, 79} (mod 120).

A139857 Primes of the form 8x^2 + 15y^2.

Original entry on oeis.org

23, 47, 167, 263, 383, 503, 647, 743, 863, 887, 983, 1103, 1223, 1367, 1487, 1583, 1607, 1823, 1847, 2063, 2087, 2207, 2423, 2447, 2543, 2663, 2687, 2903, 2927, 3023, 3167, 3407, 3527, 3623, 3767, 3863, 4007, 4127, 4463, 4583, 4703, 4943

Offset: 1

T. D. Noe, May 02 2008

Discriminant= = -480. See A139827 for more information.
Also primes of the form 12x^2 + 12xy + 23y^2, which has discriminant = -960. - T. D. Noe, May 07 2008
Also primes of the forms 23x^2 + 22xy + 47y^2 and 23x^2 + 8xy + 32y^2. See A140633. - T. D. Noe, May 19 2008

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)

[ p: p in PrimesUpTo(6000) | p mod 120 in {23, 47}]; // Vincenzo Librandi, Jul 29 2012

QuadPrimes2[8, 0, 15, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=1, sqrtint(lim\8), w=8*x^2; for(y=1, sqrtint((lim-w)\15), if(isprime(t=w+15*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 22 2017

The primes are congruent to {23, 47} (mod 120).

A139858 Primes of the form 8x^2+8xy+17y^2.

Original entry on oeis.org

17, 113, 137, 233, 257, 353, 593, 617, 857, 953, 977, 1097, 1193, 1217, 1433, 1553, 1697, 1913, 2153, 2273, 2297, 2393, 2417, 2633, 2657, 2753, 2777, 2897, 3137, 3257, 3593, 3617, 3833, 4073, 4217, 4337, 4457, 4673, 4793, 4817, 4937, 5153

Offset: 1

T. D. Noe, May 02 2008

Discriminant=-480. See A139827 for more information.
Also primes of the form 17x^2+14xy+17y^2, which has discriminant=-960. - T. D. Noe, May 07 2008
Also primes of the forms 17x^2+16xy+32y^2 and 17x^2+6xy+57y^2. See A140633. - T. D. Noe, May 19 2008

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 Item 15 of Table II.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

[ p: p in PrimesUpTo(6000) | p mod 120 in {17, 113}]; // Vincenzo Librandi, Jul 29 2012

QuadPrimes2[8, -8, 17, 10000] (* see A106856 *)

The primes are congruent to {17, 113} (mod 120).

A033202 Primes of form x^2+93*y^2.

Original entry on oeis.org

97, 109, 157, 193, 349, 373, 397, 421, 541, 577, 661, 733, 769, 853, 877, 937, 997, 1033, 1093, 1117, 1213, 1237, 1249, 1321, 1489, 1597, 1609, 1621, 1657, 1693, 1741, 1777, 1861, 1993, 2017, 2029, 2053, 2113, 2221, 2281, 2341, 2389, 2437, 2521, 2593, 2713, 2797, 2857, 2953

Offset: 1

N. J. A. Sloane

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

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 372 in {1,25,49,97, 109,121,133,157,169,193,205,253,289,349,361}]; // Vincenzo Librandi, Jul 02 2016

[p: p in PrimesUpTo(3000) | NormEquation(93,p) eq true]; // Bruno Berselli, Jul 03 2016

QuadPrimes2[1, 0, 93, 10000] (* see A106856 *)
Select[Prime@Range[500], MemberQ[{1, 25, 49, 97, 109, 121, 133, 157, 169, 193, 205, 253, 289, 349, 361}, Mod[#, 372]] &] (* Vincenzo Librandi, Jul 02 2016 *)

The primes are congruent to {1, 25, 49, 97, 109, 121, 133, 157, 169, 193, 205, 253, 289, 349, 361} (mod 372). - T. D. Noe, Apr 29 2008

A033206 Primes of form x^2+95*y^2.

Original entry on oeis.org

131, 239, 389, 419, 461, 821, 859, 919, 1051, 1109, 1531, 1601, 1879, 1901, 2011, 2399, 2411, 2609, 2699, 2791, 2971, 3011, 3041, 3469, 3541, 3559, 3671, 3709, 4139, 4219, 4261, 4349, 4451, 4679, 4691

Offset: 1

N. J. A. Sloane

Also primes of the form x^2-xy+24y^2 with x and y nonnegative. - T. D. Noe, May 08 2005

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

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[1, 0, 95, 10000] (* see A106856 *)

cp's OEIS Frontend

A106864 Primes of the form 2x^2+2xy+3y^2 with x and y nonnegative.

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A106878 Primes of the form 3x^2+xy+3y^2.

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A106889 Primes of the form 2x^2 + 5y^2.

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A106950 Primes of the form x^2 + 18y^2.

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

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A139855 Primes of the form 4x^2+4xy+31y^2.

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A139857 Primes of the form 8x^2 + 15y^2.

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A139858 Primes of the form 8x^2+8xy+17y^2.

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