A102273 -id:A102273 - cp's OEIS frontend

A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

2, 11, 23, 37, 43, 53, 71, 79, 107, 109, 127, 137, 149, 151, 163, 193, 197, 211, 233, 239, 263, 281, 317, 331, 337, 373, 389, 401, 421, 431, 443, 463, 487, 491, 499, 541, 547, 557, 569, 599, 613, 617, 641, 653, 659, 673, 683, 739, 743, 751, 757, 809, 821

Offset: 1

T. D. Noe, May 09 2005, Apr 28 2008

Discriminant=-7. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac.

Consider sequences of primes produced by forms with -100

The Mathematica function QuadPrimes2 is useful for finding the primes less than "lim" represented by the positive definite quadratic form ax^2 + bxy + cy^2 for any a, b and c satisfying a>0, c>0, and discriminant d<0. It does this by examining all x>=0 and y>=0 in the ellipse ax^2 + bxy + cy^2 <= lim. To find the primes generated by positive and negative x and y, compute the union of QuadPrimes2[a,b,c,lim] and QuadPrimes2[a,-b,c,lim]. - T. D. Noe, Sep 01 2009

For other programs see the "Binary Quadratic Forms and OEIS" link.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
L. E. Dickson, History of the Theory of Numbers, Vol. 3, Chelsea, 1923.

Zak Seidov and N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (The first 1225 terms were found by Zak Seidov)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Discriminants in the range -3 to -100: A007645 (d=-3), A002313 (d=-4), A045373, A106856 (d=-7), A033203 (d=-8), A056874, A106857 (d=-11), A002476 (d=-12), A033212, A106858-A106861 (d=-15), A002144, A002313 (d=-16), A106862-A106863 (d=-19), A033205, A106864-A106865 (d=-20), A106866-A106869 (d=-23), A033199, A084865 (d=-24), A002476, A106870 (d=-27), A033207 (d=-28), A033221, A106871-A106874 (d=-31), A007519, A007520, A106875-A106876 (d=-32), A106877-A106881 (d=-35), A040117, A068228, A106882 (d=-36), A033227, A106883-A106888 (d=-39), A033201, A106889 (d=-40), A106890-A106891 (d=-43), A033209, A106282, A106892-A106893 (d=-44), A033232, A106894-A106900 (d=-47), A068229 (d=-48), A106901-A106904 (d=-51), A033210, A106905-A106906 (d=-52), A033235, A106907-A106913 (d=-55), A033211, A106914-A106917 (d=-56), A106918-A106922 (d=-59), A033212, A106859 (d=-60), A106923-A106930 (d=-63), A007521, A106931 (d=-64), A106932-A106933 (d=-67), A033213, A106934-A106938 (d=-68), A033246, A106939-A106948 (d=-71), A106949-A106950 (d=-72), A033212, A106951-A106952 (d=-75), A033214, A106953-A106955 (d=-76), A033251, A106956-A106962 (d=-79), A047650, A106963-A106965 (d=-80), A106966-A106970 (d=-83), A033215, A102271, A102273, A106971-A106974 (d=-84), A033256, A106975-A106983 (d=-87), A033216, A106984 (d=-88), A106985-A106989 (d=-91), A033217 (d=-92), A033206, A106990-A107001 (d=-95), A107002-A107008 (d=-96), A107009-A107013 (d=-99).
Other collections of quadratic forms: A139643, A139827.
For a more comprehensive list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Cf. also A242660.

QuadPrimes2[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax}, d = b^2 - 4a*c; If[a > 0 && c > 0 && d < 0, xMax = Sqrt[lmt/a]*(1+Abs[b]/Floor[Sqrt[-d]])]; Do[ If[ 4c*lmt + d*x^2 >= 0, yMax = ((-b)*x + Sqrt[4c*lmt + d*x^2])/(2c), yMax = 0 ]; Do[p = a*x^2 + b*x*y + c*y^2; If[ PrimeQ[ p]  && p <= lmt && !MemberQ[ lst, p], AppendTo[ lst, p]], {y, 0, yMax}], {x, 0, xMax}]; Sort[ lst]];
QuadPrimes2[1, 1, 2, 1000]
(This is a corrected version of the old, incorrect, program QuadPrimes. - N. J. A. Sloane, Jun 15 2014)
max = 1000; Table[yy = {y, 1, Floor[Sqrt[8 max - 7 x^2]/4 - x/4]}; Table[ x^2 + x y + 2 y^2, yy // Evaluate], {x, 0, Floor[Sqrt[max]]}] // Flatten // Union // Select[#, PrimeQ]& (* Jean-François Alcover, Oct 04 2018 *)

list(lim)=my(q=Qfb(1,1,2), v=List([2])); forprime(p=2, lim, if(vecmin(qfbsolve(q, p))>0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Aug 05 2016

Removed old Mathematica programs - T. D. Noe, Sep 09 2009

Edited (pointed out error in QuadPrimes, added new version of program, checked and extended b-file). - N. J. A. Sloane, Jun 06 2014

A140633 Primes of the form 7x^2+4xy+52y^2.

Original entry on oeis.org

7, 103, 127, 223, 367, 463, 487, 607, 727, 823, 967, 1063, 1087, 1303, 1327, 1423, 1447, 1543, 1567, 1663, 1783, 2143, 2287, 2383, 2503, 2647, 2767, 2887, 3343, 3463, 3583, 3607, 3727, 3823, 3847, 3943, 3967, 4327, 4423, 4447, 4567, 4663

Offset: 1

T. D. Noe, May 19 2008

Discriminant=-1440. Also primes of the forms 7x^2+6xy+87y^2 and 7x^2+2xy+103y^2.
Voight proves that there are exactly 69 equivalence classes of positive definite binary quadratic forms that represent almost the same primes. 48 of those quadratic forms are of the idoneal type discussed in A139827. The remaining 21 begin at A140613 and end here. The cross-references section lists the quadratic forms in the same order as tables 1-6 in Voight's paper. Note that A107169 and A139831 are in the same equivalence class.
In base 12, the sequence is 7, 87, X7, 167, 267, 327, 347, 427, 507, 587, 687, 747, 767, 907, 927, 9X7, X07, X87, XX7, E67, 1047, 12X7, 13X7, 1467, 1547, 1647, 1727, 1807, 1E27, 2007, 20X7, 2107, 21X7, 2267, 2287, 2347, 2367, 2607, 2687, 26X7, 2787, 2847, where X is for 10 and E is for 11. Moreover, the discriminant is X00 and that all elements are {7, 87, X7, 167, 187, 247} mod 260. - Walter Kehowski, May 31 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)
John Voight, Quadratic forms that represent almost the same primes, Math. Comp., Vol. 76 (2007), pp. 1589-1617.

Cf. A033205, A007519, A068228, A107135, A107144, A107152, A107151.
Cf. A107181, A139502, A139856, A139854, A139874, A139877, A139897.
Cf. A140613, A140614. A140615, A139923, A140616-A140619, A139988.
Cf. A139993, A140008, A140010, A140620-A140632, A033212, A102273.
Cf. A107007, A107003, A139830, A107169, A139831, A141373, A139847.
Cf. A139850, A139855, A139860, A139879, A139880, A139924, A139990.
Cf. A139998, A139992, A139996, A140003, A140013, A107006, A139858.
Cf. A107008, A140633, A139991, A139857, A139859, A139878, A007645, A002313.

Union[QuadPrimes2[7, 4, 52, 10000], QuadPrimes2[7, -4, 52, 10000]] (* see A106856 *)

A102269 Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = +1, chi_{-7} = +1.

Original entry on oeis.org

43, 67, 79, 127, 151, 163, 211, 331, 379, 463, 487, 499, 547, 571, 631, 739, 751, 823, 883, 907, 919, 967, 991, 1051, 1087, 1171, 1303, 1327, 1423, 1471, 1579, 1663, 1723, 1747, 1759, 1831, 1999, 2011, 2083, 2143, 2179, 2251, 2311, 2347, 2503, 2647, 2671, 2683, 2731, 2767, 2851, 3019

Offset: 1

N. J. A. Sloane, Feb 19 2005

nonn

Primes p such that p is 3 (mod 4) and (-3/p) = (-7/p) = 1, where (k/n) is the Kronecker symbol. - Robin Visser, Mar 13 2024

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.

Cf. A102270, A102271, A102272, A102273, A102274, A102275.

[p : p in PrimesUpTo(3000) | p mod 84 in [43, 67, 79]];  // Robin Visser, Mar 13 2024

The primes are congruent to {43, 67, 79} (mod 84). - Robin Visser, Mar 13 2024

More terms from Robin Visser, Mar 13 2024

A102275 2-class number of Q(sqrt(-21p)) as p runs through primes in A102274.

Original entry on oeis.org

8, 32, 8, 8, 8, 32, 64, 8, 16, 16, 64, 8, 16, 16, 16, 8, 32, 8, 8, 8, 32, 8, 16, 8, 8, 8, 8, 8, 32, 16, 8, 16, 128, 32, 32, 8, 32, 16, 32, 8, 128, 16, 16, 16, 8, 8, 16, 8, 16, 8, 8, 8, 16, 16, 16, 8, 64, 8, 8, 16, 8, 8, 8, 8, 8, 8, 8, 16, 16, 16, 32, 16, 8, 16, 32, 16, 8, 8, 32, 16, 16

Offset: 1

N. J. A. Sloane, Feb 19 2005

nonn

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.

Cf. A102269, A102270, A102271, A102272, A102273, A102274.

[2^QuadraticField(-21*p).class_number().valuation(2) for p in Primes()[:1000] if (p%84) in [47, 59, 83]] # Robin Visser, Mar 13 2024

a(19) corrected and more terms from Robin Visser, Mar 13 2024

A102270 2-class number of Q(sqrt(-21p)) as p runs through primes in A102269.

Original entry on oeis.org

16, 8, 8, 8, 16, 8, 16, 16, 16, 8, 16, 32, 8, 8, 8, 8, 32, 8, 16, 64, 32, 8, 32, 8, 8, 128, 32, 32, 32, 16, 8, 8, 8, 16, 16, 8, 8, 8, 128, 16, 8, 8, 16, 32, 8, 16, 32, 8, 8, 8, 128, 8, 8, 8, 16, 8, 64, 8, 8, 8, 16, 8, 16, 16, 16, 8, 8, 16, 32, 16, 8, 8, 8, 64, 32, 64, 16, 64, 8, 16, 32

Offset: 1

N. J. A. Sloane, Feb 19 2005

nonn

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.

Cf. A102269, A102271, A102272, A102273, A102274, A102275.

[2^QuadraticField(-21*p).class_number().valuation(2) for p in Primes()[:1000] if (p%84) in [43, 67, 79]] # Robin Visser, Mar 13 2024

More terms from Robin Visser, Mar 13 2024

A102272 2-class number of Q(sqrt(-21p)) as p runs through primes in A102271.

Original entry on oeis.org

16, 8, 8, 8, 16, 16, 8, 32, 16, 16, 16, 8, 16, 8, 64, 8, 8, 8, 8, 128, 16, 8, 8, 32

Offset: 1

N. J. A. Sloane, Feb 19 2005

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.

Cf. A102269, A102270, A102271, A102273, A102274, A102275.

A102274 Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = -1, chi_{-7} = -1.

Original entry on oeis.org

47, 59, 83, 131, 167, 227, 251, 311, 383, 419, 467, 479, 503, 563, 587, 647, 719, 839, 887, 971, 983, 1091, 1151, 1223, 1259, 1307, 1319, 1427, 1487, 1511, 1559, 1571, 1811, 1823, 1847, 1907, 1931, 1979, 2063, 2099, 2243, 2267, 2351, 2399, 2411, 2579, 2663, 2687, 2819, 2903, 2939, 2999

Offset: 1

N. J. A. Sloane Feb 19 2005

nonn

Primes p such that p is 3 (mod 4) and (-3/p) = (-7/p) = -1, where (k/n) is the Kronecker symbol. - Robin Visser, Mar 13 2024

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.

Cf. A102269, A102270, A102271, A102272, A102273, A102275.

[p : p in PrimesUpTo(3000) | p mod 84 in [47, 59, 83]];  // Robin Visser, Mar 13 2024

The primes are congruent to {47, 59, 83} (mod 84). - Robin Visser, Mar 13 2024

More terms from Robin Visser, Mar 13 2024

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cp's OEIS Frontend

A106856 Primes of the form x^2 + xy + 2y^2, with x and y nonnegative.

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A140633 Primes of the form 7x^2+4xy+52y^2.

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Mathematica

A102269 Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = +1, chi_{-7} = +1.

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A102275 2-class number of Q(sqrt(-21p)) as p runs through primes in A102274.

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A102270 2-class number of Q(sqrt(-21p)) as p runs through primes in A102269.

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Sage

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A102272 2-class number of Q(sqrt(-21p)) as p runs through primes in A102271.

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A102274 Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = -1, chi_{-7} = -1.

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Magma

Formula

Extensions