A105389 -id:A105389 - cp's OEIS frontend

A107132 Primes of the form 2x^2 + 13y^2.

2, 13, 31, 149, 167, 317, 359, 397, 463, 487, 509, 613, 661, 709, 839, 1061, 1087, 1103, 1151, 1181, 1367, 1471, 1783, 1789, 1861, 2039, 2111, 2221, 2269, 2437, 2503, 2621, 2647, 2917, 2927, 2957, 3023, 3079, 3167, 3229, 3373, 3541, 3853

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -104. Binary quadratic forms ax^2+cy^2 have discriminant d=-4ac. We consider sequences of primes produced by forms with -400<=d<=0, a<=c and gcd(a,c)=1. These restrictions yield 173 sequences of prime numbers, which are organized by discriminant below. See A106856 for primes of the form ax^2+bxy+cy^2 with discriminant > -100.

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.

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. A033218 (d=-104), A014752 (d=-108), A107133, A107134 (d=-112), A033219 (d=-116), A107135-A107137, A033220 (d=-120), A033221 (d=-124), A105389 (d=-128), A107138, A033222 (d=-132), A107139, A033223 (d=-136), A107140, A033224 (d=-140), A107141, A107142 (d=-144), A033225 (d=-148), A107143, A033226 (d=-152), A033227 (d=-156), A107144, A107145 (d=-160), A033228 (d=-164), A107146-A107148, A033229 (d=-168).
Cf. A033230 (d=-172), A107149, A107150 (d=-176), A107151, A107152 (d=-180), A107153, A033231 (d=-184), A033232 (d=-188), A141373 (d=-192), A107155 (d=-196), A107156, A107157 (d=-200), A107158, A033233 (d=-204), A107159, A107160 (d=-208), A033234 (d=-212), A107161, A107162 (d=-216), A033235 (d=-220), A107163, A107164 (d=-224), A107165, A033236 (d=-228), A107166, A033237 (d=-232), A033238 (d=-236).
Cf. A107167-A107169 (d=-240), A033239 (d=-244), A107170, A033240 (d=-248), A014754 (d=-256), A107171, A033241 (d=-260), A107172-A107174, A033242 (d=-264), A033243 (d=-268), A107175, A107176 (d=-272), A107177, A033244 (d=-276), A107178-A107180, A033245 (d=-280), A033246 (d=-284), A107181 (d=-288), A033247 (d=-292), A107182, A033248 (d=-296), A107183, A107184 (d=-300), A107185, A107186 (d=-304), A107187, A033249 (d=-308).
Cf. A107188-A107190, A033250 (d=-312), A033251 (d=-316), A107191, A107192 (d=-320), A107193 (d=-324), A107194, A033252 (d=-328), A033253 (d=-332), A107195-A107198 (d=-336), A107199, A033254 (d=-340), A107200, A033255 (d=-344), A033256 (d=-348), A107132 A107201, A107202 (d=-352), A033257 (d=-356), A107203-A107206 (d=-360), A107207, A033258 (d=-364), A107208, A107209 (d=-368), A107210, A033202 (d=-372).
Cf. A107211, A033204 (d=-376), A033206 (d=-380), A107212, A107213 (d=-384), A033208 (d=-388), A107214, A107215 (d=-392), A107216, A107217 (d=-396), A107218, A107219 (d=-400).
For a more complete list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

QuadPrimes2[2, 0, 13, 10000] (* see A106856 *)

list(lim)=my(v=List([2,13]),t); for(y=1,sqrtint(lim\13), for(x=1,sqrtint((lim-13*y^2)\2), if(isprime(t=2*x^2+13*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 07 2017

A121243 Primes of the form 4x^2 + 4xy + 9y^2.

Original entry on oeis.org

17, 73, 89, 97, 193, 233, 241, 281, 401, 433, 449, 601, 617, 641, 673, 769, 929, 937, 977, 1009, 1033, 1049, 1097, 1193, 1289, 1297, 1361, 1409, 1433, 1481, 1489, 1609, 1697, 1721, 1753, 1801, 1873, 1913

Offset: 1

Steven Finch, Aug 22 2006

nonn

This sequence is complementary to A105389 in the sense that the two sequences are disjoint and their union constitutes all primes p satisfying Mod[p,8]=1.

Primes satisfying Mod[p,8]=1 are of form x^2+8y^2 (A007519), with the sequence above as odd y, while A105389 is even y. This can be seen by expressing the former as (2x+y)^2+8y^2 (where y can only be odd), while the latter is u^2+8(2v)^2. [From Tito Piezas III, Jan 01 2009]

			17 = 4*1^2 + 4*1*1 + 9*1^2, 73 = 4*1^2 + 4*1*(-3) + 9*(-3)^2

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/0701251 [math.NT], 2007.
Claudio Qureshi, Antonio Campello, Sueli I. R. Costa, Non-Existence of Linear Perfect Lee Codes With Radius 2 for Infinitely Many Dimensions, IEEE Transactions on Information Theory (2018) Vol. 64, Issue 4, pp. 3042-3047.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A105389.

QuadPrimes2[4, -4, 9, 10000] (* see A106856 *)
(* Second program: *)
max = 2000; Table[yy = {y, Floor[-2x/9 - 1/9 Sqrt[9max - 32x^2]], Ceiling[-2x/9 + 1/9 Sqrt[9max - 32x^2]]}; Table[4x^2 + 4 x y + 9y^2, yy // Evaluate], {x, 0, Ceiling[3Sqrt[max]/(4Sqrt[2])]}] // Flatten // Union // Select[#, # <= max && PrimeQ[#]&]& // Quiet (* Jean-François Alcover, Oct 08 2018 *)

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A107132 Primes of the form 2x^2 + 13y^2.

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A121243 Primes of the form 4x^2 + 4xy + 9y^2.

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cp's OEIS Frontend

A107132 Primes of the form 2x^2 + 13y^2.

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Author

Keywords

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Crossrefs

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PARI

A121243 Primes of the form 4*x^2 + 4*x*y + 9*y^2.

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A121243 Primes of the form 4x^2 + 4xy + 9y^2.