A107145 -id:A107145 - 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

A139490 Numbers n such that the quadratic form x^2 + nxy + y^2 represents exactly the same primes as the quadratic form x^2 + m*y^2 for some m.

Original entry on oeis.org

1, 4, 6, 7, 8, 10, 14, 16, 18, 22, 26, 38, 58, 82, 86

Offset: 1

Artur Jasinski, Apr 24 2008, Apr 26 2008, Apr 27 2008

nonn

For the numbers m see A139491.
Conjecture: This sequence is finite and complete (checked for range n<=200 and m<=500).
Three more terms were found by searching n <= 1000 and m <= 4000. The corresponding m are 840, 840, and 1848, which are idoneal numbers A000926. The sequence is probably complete now. [T. D. Noe, Apr 27 2009]

			a(1)=1 because the primes represented by x^2+xy+y^2 are the same as the primes represented by x^2 + 3*y^2 (see A007645).
The known pairs (n,m) are the following (checked for range n<=200 and m<=500):
n={1, 4, 4, 6, 6, 7, 8, 8, 10, 10, 10, 14, 14, 14, 16, 18, 22, 22, 26, 38, 38}
m={3, 9, 12, 8, 16, 15, 45, 60, 24, 48, 72, 24, 48, 72, 21, 40, 120, 240, 168, 120, 240}.

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, A139491.

f = 200; g = 300; h = 30; j = 100; b = {}; Do[a = {}; Do[Do[If[PrimeQ[x^2 + n y^2], AppendTo[a, x^2 + n y^2]], {x, 0, g}], {y, 1, g}]; AppendTo[b, Take[Union[a], h]], {n, 1, f}]; Print[b]; c = {}; Do[a = {}; Do[Do[If[PrimeQ[n^2 + w*n*m + m^2], AppendTo[a, n^2 + w*n*m + m^2]], {n, m, g}], {m, 1, g}]; AppendTo[c, Take[Union[a], h]], {w, 1, j}]; Print[c]; bb = {}; cc = {}; Do[Do[If[b[[p]] == c[[q]], AppendTo[bb, p]; AppendTo[cc, q]], {p, 1, f}], {q, 1, j}]; Union[cc] (*Artur Jasinski*)

Edited by N. J. A. Sloane, Apr 25 2008
Extended by T. D. Noe, Apr 27 2009
Typo fixed by Charles R Greathouse IV, Oct 28 2009

A139492 Primes of the form x^2 + 5x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

7, 37, 43, 67, 79, 109, 127, 151, 163, 193, 211, 277, 331, 337, 373, 379, 421, 457, 463, 487, 499, 541, 547, 571, 613, 631, 673, 709, 739, 751, 757, 823, 877, 883, 907, 919, 967, 991, 1009, 1033, 1051, 1087, 1093, 1117, 1129, 1171, 1201, 1213, 1297, 1303

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Reduced form is [1, 3, -3]. Discriminant = 21. Class number = 2.
Values of the quadratic form are {0, 1, 3, 4} mod 6, so this is a subsequence of A002476. - R. J. Mathar, Jul 30 2008
It can be checked that the primes p of the form x^2 + n*x*y + y^2, n >= 3, where x and y are nonnegative, depend on n mod 6 as follows: n mod 6 = 0 => p mod 12 = {1,5}; n mod 6 = 1 => p mod 12 = {1,7}; n mod 6 = 2 => p mod 12 = {1}; n mod 6 = 3 => p mod 12 = {1,5,7,11}; n mod 6 = 4 => p mod 12 = {1}; n mod 6 = 5 => p mod 12 = {1,7}. - Walter Kehowski, Jun 01 2008

			a(1) = 7 because we can write 7 = 1^2 + 5*1*1 + 1^2.

Z. I. Borevich and I. R. Shafarevich, Number Theory.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

Peter Luschny, Binary Quadratic Forms
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.

Primes in A243172.
Cf. A002476, A007645, A007519, A033212, A033215, A068228, A107008, A107145, A107152, A139490, A139489, A139491, A139492, A141159.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

a = {}; w = 5; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a]

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 5, 1])
print(Q.represented_positives(1303, 'prime')) # Peter Luschny, May 12 2021

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

A139494 Primes of the form x^2 + 11x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

13, 43, 61, 79, 103, 127, 139, 157, 181, 199, 211, 277, 283, 313, 337, 367, 373, 433, 439, 523, 547, 571, 601, 607, 673, 727, 751, 757, 823, 829, 859, 883, 907, 919, 937, 991, 997, 1039, 1063, 1069, 1093, 1117, 1153, 1171, 1213, 1231, 1249, 1291, 1297

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

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.

a = {}; w = 11; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

A139506 Primes of the form x^2 + 26x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

193, 337, 457, 673, 1009, 1033, 1129, 1201, 1297, 1801, 1873, 2017, 2137, 2377, 2473, 2521, 2689, 2713, 2857, 3049, 3217, 3313, 3361, 3529, 3697, 3889, 4057, 4153, 4201, 4561, 4657, 4729, 4993, 5209, 5233, 5569, 5737, 5881, 6073, 6217, 6337, 6553, 6577

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Also primes of the form x^2 + 168y^2. - T. D. Noe, Apr 29 2008

In base 12, the sequence is 141, 241, 321, 481, 701, 721, 7X1, 841, 901, 1061, 1101, 1201, 12X1, 1461, 1521, 1561, 1681, 16X1, 17X1, 1921, 1X41, 1E01, 1E41, 2061, 2181, 2301, 2421, 24X1, 2521, 2781, 2841, 28X1, 2X81, 3021, 3041, 3281, 33X1, 34X1, 3621, 3721, 3801, 3961, 3981, where X is 10 and E is 11. Moreover, the discriminant is 480. - Walter Kehowski, Jun 01 2008

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

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

Cf. A139643.

a = {}; w = 26; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a]

The primes are congruent to {1, 25, 121} (mod 168). - T. D. Noe, Apr 29 2008

A139512 Primes of the form x^2 + 32xy + y^2 for x and y nonnegative.

Original entry on oeis.org

229, 349, 409, 421, 661, 769, 829, 1021, 1069, 1249, 1381, 1429, 1549, 1789, 1801, 1861, 2089, 2161, 2269, 2389, 3001, 3061, 3109, 3181, 3229, 3469, 3889, 4021, 4129, 4201, 4441, 4861, 4909, 5101, 5449, 5521, 5869, 5881, 6121, 6469, 6481, 6529, 6781

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Are all terms == 1 mod 12? - Zak Seidov, Apr 25 2008

Yes: (i) all terms == 1 mod 3 because the quadratic form has terms == {0,1} mod 3 and the values ==0 mod 3 are not primes. (ii) all terms == 1 mod 4 because the quadratic form has terms == {0,1,2} mod 4 and the values = {0,2} mod 4 are not primes. By the Chinese remainder constructions for coprime 3 and 4 all prime terms are == 1 mod 12. - R. J. Mathar, Jun 10 2020

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references), discriminant 1020.

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

a = {}; w = 32; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

A155488 Primes p with property that p^2 is of the form x^2 + 40y^2.

Original entry on oeis.org

7, 11, 13, 19, 23, 37, 41, 47, 53, 59, 89, 103, 127, 131, 139, 157, 167, 173, 179, 197, 211, 223, 241, 251, 263, 277, 281, 293, 317, 331, 367, 373, 379, 383, 397, 401, 409, 419, 449, 463, 487, 491, 499, 503, 521, 557, 569, 571, 601, 607, 613, 619, 641, 647

Offset: 1

Zak Seidov, Jan 23 2009

nonn

All p^2 are congruent to {1, 9} (mod 40), as in A107145.

Rational primes that decompose in the field Q(sqrt(-10)). - N. J. A. Sloane, Dec 26 2017

Robert Price, Table of n, a(n) for n = 1..1000
Zak Seidov, First 1000 terms
Index to sequences related to decomposition of primes in quadratic fields

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

Load the Maple program HH given in A296920. Then run HH(-10, 200); This produces A155488, A296925, A293859. - N. J. A. Sloane, Dec 26 2017

Sqrt[#]&/@Select[Union[Flatten[Table[x^2+40y^2,{x,800},{y,100}]]],PrimeQ[Sqrt[#]]&] (* Harvey P. Dale, Jul 12 2025 *)

A139505 Primes of the form x^2 + 25x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

151, 163, 307, 397, 409, 541, 547, 601, 673, 811, 823, 859, 967, 997, 1153, 1231, 1237, 1327, 1567, 1669, 1741, 1879, 2083, 2143, 2281, 2293, 2557, 2677, 2707, 2833, 2971, 3037, 3259, 3313, 3433, 3877, 4003, 4129, 4153, 4603, 4639, 4861, 4957, 5101, 5227

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..5000
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.

a = {}; w = 25; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)
With[{nn=80},Select[Union[#[[1]]^2+25#[[1]]#[[2]]+#[[2]]^2&/@Tuples[ Range[ 0,nn],2]],PrimeQ[#]&&#Harvey P. Dale, Feb 10 2020 *)

A139493 Primes of the form x^2 + 9x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

11, 23, 37, 53, 67, 71, 113, 137, 163, 179, 191, 317, 331, 379, 389, 401, 421, 443, 449, 463, 487, 499, 599, 617, 631, 641, 653, 683, 709, 751, 757, 823, 863, 883, 907, 911, 947, 977, 991, 1061, 1087, 1093, 1103, 1171, 1213, 1303, 1367, 1373, 1409, 1423

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

This is a member of the family of sequences of primes of the forms x^2 + kxy + y^2.

See for k=1 A007645 = x^2+3y^2, k=2 squares no primes, k=3 A038872, k=4 A068228 = x^2+9y^2, k=5 A139492, k=6 A007519 = x^2+8y^2, k=7 A033212 = x^2+15y^2, k=8 A107152 = x^2+45y^2, k=9 A139493, k=10 A107008 = x^2+24y^2, k=11 A139494, k=12 A139495, k=13 A139496, k=14* = 10 A107008 = x^2+24y^2, k=15 A139497, k=16 A033215 = x^2+21y^2, k=17 A139498, k=18 A107145 = x^2+40y^2, k=19 A139499, k=20 A139500, k=21 A139501, k=22 A139502, k=23 A139503, k=24 A139504, k=25 A139505, k=26,A139506, k=27 A139507, k=28 A139508, k=29 A139509, k=30 A139510, k=31 A139511, k=32 A139512

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.

a = {}; w = 9; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

cp's OEIS Frontend

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

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A139490 Numbers n such that the quadratic form x^2 + n*x*y + y^2 represents exactly the same primes as the quadratic form x^2 + m*y^2 for some m.

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A139492 Primes of the form x^2 + 5x*y + y^2 for x and y nonnegative.

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

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A139494 Primes of the form x^2 + 11x*y + y^2 for x and y nonnegative.

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A139506 Primes of the form x^2 + 26x*y + y^2 for x and y nonnegative.

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A139512 Primes of the form x^2 + 32*x*y + y^2 for x and y nonnegative.

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A155488 Primes p with property that p^2 is of the form x^2 + 40y^2.

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A139490 Numbers n such that the quadratic form x^2 + nxy + y^2 represents exactly the same primes as the quadratic form x^2 + m*y^2 for some m.

A139512 Primes of the form x^2 + 32xy + y^2 for x and y nonnegative.