A140633 -id:A140633 - cp's OEIS frontend

A102273 Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = -1, chi_{-7} = +1.

11, 23, 71, 107, 179, 191, 239, 263, 347, 359, 431, 443, 491, 599, 659, 683, 743, 827, 863, 911, 947, 1019, 1031, 1103, 1163, 1187, 1283, 1367, 1439, 1451, 1499, 1523, 1583, 1607, 1619, 1667, 1787, 1871, 2003, 2027, 2039, 2087

Offset: 1

N. J. A. Sloane, Feb 19 2005

nonn

The 2-class number of these fields is always 4.
Primes of the form 2x^2 - 2xy + 11y^2 with x nonnegative and y positive. - T. D. Noe, May 08 2005
Also primes of the forms 8x^2 + 4xy + 11y^2 and 11x^2 + 2xy + 23y^2. See A140633. - T. D. Noe, May 19 2008
The discriminant of positive definite binary quadratic form (2,2,11) is -84. - Hugo Pfoertner, Jul 14 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
H. Cohn and J. C. Lagarias, On the existence of fields governing the 2-invariants of the classgroup of Q(sqrt{dp}) as p varies, Math. Comp. 41 (1983), 711-730.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A102269-A102275.

Cf. A139827.

[p: p in PrimesUpTo(3000) | p mod 84 in [2, 11, 23, 71]]; // Vincenzo Librandi, Jul 19 2012

f[x_,y_]:=2*x^2+2*x*y+11*y^2; lst={};Do[Do[p=f[x,y];If[PrimeQ[p],AppendTo[lst,p]],{y,-5!,6!}],{x,-5!,6!}];Take[Union[lst],5! ] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2009 *)

The primes are congruent to {2, 11, 23, 71} (mod 84). - T. D. Noe, May 02 2008

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

A107007 Primes of the form 3x^2+8y^2.

Original entry on oeis.org

3, 11, 59, 83, 107, 131, 179, 227, 251, 347, 419, 443, 467, 491, 563, 587, 659, 683, 827, 947, 971, 1019, 1091, 1163, 1187, 1259, 1283, 1307, 1427, 1451, 1499, 1523, 1571, 1619, 1667, 1787, 1811, 1907, 1931, 1979, 2003, 2027, 2099, 2243, 2267

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-96.
Except for 3, also primes of the forms 8*x^2+8*x*y+11*y^2 and 11*x^2+6*x*y+27*y^2. See A140633. - T. D. Noe, May 19 2008
Except for the first member, 3, all the members seem to be terms of A123239 which are prime in both k(i) and k(rho). - A.K. Devaraj, Nov 24 2009

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 24 in {11} ]; // Vincenzo Librandi, Jul 23 2012

QuadPrimes2[3, 0, 8, 10000] (* see A106856 *)

Except for 3, the terms are congruent to 11 (mod 24). - T. D. Noe, May 02 2008

A139502 Primes of the form x^2 + 22x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

241, 409, 601, 769, 1009, 1129, 1201, 1249, 1321, 1489, 1609, 1801, 2089, 2161, 2281, 2521, 2689, 3001, 3049, 3121, 3169, 3361, 3529, 3769, 3889, 4129, 4201, 4441, 4561, 4729, 4801, 4969, 5209, 5281, 5449, 5521, 5569, 5641, 5689, 5881, 6121, 6361, 6481

Offset: 1

Artur Jasinski, Apr 24 2008

Also primes of the form x^2 + 120y^2. - T. D. Noe, Apr 29 2008
Also primes of the form x^2+240y^2. See A140633. - T. D. Noe, May 19 2008
In base 12, the sequence is 181, 2X1, 421, 541, 701, 7X1, 841, 881, 921, X41, E21, 1061, 1261, 1301, 13X1, 1561, 1681, 18X1, 1921, 1981, 1X01, 1E41, 2061, 2221, 2301, 2481, 2521, 26X1, 2781, 28X1, 2941, 2X61, 3021, 3081, 31X1, 3241, 3281, 3321, 3361, 34X1, 3661, 3821, 3901, where X is 10 and E is 11. Moreover, the discriminant is 340. - Walter Kehowski, Jun 01 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

Cf. A139643.

[ p: p in PrimesUpTo(7000) | p mod 120 in {1, 49}]; // Vincenzo Librandi, Jul 28 2012

QuadPrimes2[1, 0, 120, 10000] (* see A106856 *)

The primes are congruent to {1, 49} (mod 120). - T. D. Noe, Apr 29 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

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).

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

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

cp's OEIS Frontend

A102273 Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = -1, chi_{-7} = +1.

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

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A107007 Primes of the form 3*x^2+8*y^2.

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A139502 Primes of the form x^2 + 22x*y + y^2 for x and y nonnegative.

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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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Magma

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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.

A107007 Primes of the form 3x^2+8y^2.