A068228 -id:A068228 - cp's OEIS frontend

A141123 Primes of the form -x^2+2xy+2y^2 (as well as of the form 3x^2+6xy+2*y^2).

2, 3, 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263, 311, 347, 359, 383, 419, 431, 443, 467, 479, 491, 503, 563, 587, 599, 647, 659, 683, 719, 743, 827, 839, 863, 887, 911, 947, 971, 983, 1019, 1031, 1091, 1103, 1151, 1163, 1187, 1223

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 05 2008

nonn

Discriminant = 12. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.

This is exactly {2} U A068231, primes congruent to 11 (mod 12). This is because the orders of imaginary quadratic fields with discriminant 12 has 1 class per genus (can be verified by the quadclassunit() function in PARI), so the primes represented by a binary quadratic form of this discriminant are determined by a congruence condition. - Jianing Song, Jun 22 2025

			a(3) = 11 because we can write 11 = -1^2 + 2*1*2 + 2*2^2 (or 11 = 3*1^2 + 6*1*1 + 2*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Robert Israel, Table of n, a(n) for n = 1..6514
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A038872 (d=5), A038873 (d=8), A068228 (d=12, 48, or -36), A038883 (d=13), A038889 (d=17), A141111 and A141112 (d=65).
Essentially the same as A068231 and A141187.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Cf. A084917.

N:= 2000:
S:= NULL:
for xx from 1 to floor(2*sqrt(N/3)) do
  for yy from ceil(sqrt(max(1,3*xx^2-N))) to floor(sqrt(3)*xx) do
     S:= S, 3*xx^2-yy^2;
od od:
sort(convert(select(isprime,{S}),list)); # Robert Israel, Jul 20 2020

Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -x^2 + 2*x*y + 2*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]]
(* or: *)
Select[Prime[Range[200]], # == 2 || # == 3 || Mod[#, 12] == 11&] (* Jean-François Alcover, Oct 25 2016, updated Oct 29 2016 *)

More terms from Colin Barker, Apr 05 2015

A068231 Primes congruent to 11 mod 12.

Original entry on oeis.org

11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263, 311, 347, 359, 383, 419, 431, 443, 467, 479, 491, 503, 563, 587, 599, 647, 659, 683, 719, 743, 827, 839, 863, 887, 911, 947, 971, 983, 1019, 1031, 1091, 1103, 1151, 1163, 1187, 1223

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

Intersection of A002145 (primes of form 4n+3) and A003627 (primes of form 3n-1). So these are both Gaussian primes with no imaginary part and Eisenstein primes with no imaginary part. - Alonso del Arte, Mar 29 2007
Is this the same sequence as A141187 (apart from the initial 3)?
If p is prime of the form 2*a(n)^k + 1, then p divides a cyclotomic number Phi(a(n)^k, 2). - Arkadiusz Wesolowski, Jun 14 2013
Also a(n) = primes p dividing A014138((p-3)/2), where A014138(n) = Partial sums of (Catalan numbers starting 1,2,5,...), cf. A000108. - Alexander Adamchuk, Dec 27 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A068227, A068228, A068229, A040117, A068232, A068233, A068234, A068235, A000040, A014138, A000108.

%4n-1 and 6n-1 primes
n = 1:10000;
n2 = 4*n-1;
n3 = 3*n-1;
p = primes(max(n2));
Res = intersect(n2,n3);
Res2 = intersect(Res,p);
% Jesse H. Crotts, Sep 25 2016

[p: p in PrimesUpTo(1500) | p mod 12 eq 11 ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime/@Range[250], Mod[ #, 12]==11&]
Select[Range[11,1500,12],PrimeQ] (* Harvey P. Dale, Sep 15 2023 *)

for(i=1,250, if(prime(i)%12==11, print(prime(i))))

Edited by Dean Hickerson, Feb 27 2002

A068229 Primes congruent to 7 (mod 12).

Original entry on oeis.org

7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271, 283, 307, 331, 367, 379, 439, 463, 487, 499, 523, 547, 571, 607, 619, 631, 643, 691, 727, 739, 751, 787, 811, 823, 859, 883, 907, 919, 967, 991, 1039, 1051, 1063, 1087, 1123, 1171, 1231

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

Primes of the form 3x^2 + 4y^2. - T. D. Noe, May 08 2005

It appears that all terms starting from term 103 are primes which are the sum of 5 positive (n > 0) different squares in more than one way (A193143) - Vladimir Joseph Stephan Orlovsky, Jul 16 2011.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A068227, A068228, A040117, A068231, A068232, A068233, A068234, A068235.

[ p: p in PrimesUpTo(1400) | p mod 12 in {7} ]; // Vincenzo Librandi, Jul 14 2012

Select[Prime/@Range[250], Mod[#, 12] == 7 &]

for(i=1,250, if(prime(i)%12==7, print(prime(i))))

is_A068229(n)=n%12==7 && isprime(n) \\ then, e.g.,
select(is_A068229, primes(250))  \\ - M. F. Hasler, Jan 25 2013

a(n) ~ 4n log n. - Charles R Greathouse IV, Dec 07 2022

Edited by Dean Hickerson, Feb 27 2002

A040117 Primes congruent to 5 (mod 12). Also primes p such that x^4 = 9 has no solution mod p.

Original entry on oeis.org

5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269, 281, 293, 317, 353, 389, 401, 449, 461, 509, 521, 557, 569, 593, 617, 641, 653, 677, 701, 761, 773, 797, 809, 821, 857, 881, 929, 941, 953, 977, 1013, 1049, 1061, 1097, 1109, 1181, 1193

Offset: 1

N. J. A. Sloane

Odd primes of the form 2x^2-2xy+5y^2 with x and y nonnegative. - T. D. Noe, May 08 2005, corrected by M. F. Hasler, Jul 03 2025
Complement of A040116 relative to A000040. - Vincenzo Librandi, Sep 17 2012
Odd primes of the form a^2 + b^2 such that a^2 == b^2 (mod 3). - Thomas Ordowski and Charles R Greathouse IV, May 20 2015
Yasutoshi Kohmoto observes that nextprime(a(n)) is more frequently congruent to 3 (mod 4) than to 1. This bias can be explained by the possible prime constellations and gaps: To have the same residue mod 4 as a prime in the list, the next prime must be at a gap of 4 or 8 or 12..., but a gap of 4 is impossible because 12k + 5 + 4 is divisible by 3, and gaps >= 12 are very rare for small primes. To have the residue 3 (mod 4) the next prime can be at a gap of 2 or 6 with no a priori divisibility property. However, this bias tends to disappear as the primes (and average prime gaps) grow bigger: for primes < 10^5, the ratio is about 35% vs 65% (as the above simple explanation suggests), but considering primes up to 10^8 yields a ratio of about 40% vs 60%. It can be expected that the ratio asymptotically tends to 1:1. - M. F. Hasler, Sep 01 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A068227, A068228, A068229, A068231, A068232, A068233, A068234, A068235.

Equal to A243183 (primes of the form 2x^2+2xy+5y^2) except for the additional A243183(1) = 2 (and indexing of subsequent terms).

[p: p in PrimesUpTo(1200) | not exists{x : x in ResidueClassRing(p) | x^4 eq 9} ]; // Vincenzo Librandi, Sep 17 2012

Select[Prime/@Range[250], Mod[ #, 12]==5&]
ok[p_]:= Reduce[Mod[x^4 - 9, p] == 0, x, Integers] == False;Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 17 2012 *)

for(i=1,250, if(prime(i)%12==5, print(prime(i))))

a(n) ~ 4n log n. - Charles R Greathouse IV, May 20 2015

More terms from Dean Hickerson, Feb 27 2002

A132230 Primes congruent to 1 (mod 30).

Original entry on oeis.org

31, 61, 151, 181, 211, 241, 271, 331, 421, 541, 571, 601, 631, 661, 691, 751, 811, 991, 1021, 1051, 1171, 1201, 1231, 1291, 1321, 1381, 1471, 1531, 1621, 1741, 1801, 1831, 1861, 1951, 2011, 2131, 2161, 2221, 2251, 2281, 2311, 2341, 2371, 2521, 2551, 2671

Offset: 1

Omar E. Pol, Aug 15 2007

Also primes congruent to 1 (mod 15). - N. J. A. Sloane, Jul 11 2008

Primes ending in 1 with (SOD-1)/3 integer where SOD is sum of digits. - Ki Punches, Feb 04 2009

			From _Muniru A Asiru_, Nov 01 2017: (Start)
31 is a prime and 31 = 30*1 + 1;
61 is a prime and 61 = 30*2 + 1;
151 is a prime and 151 = 30*5 + 1;
211 is a prime and 211 = 30*7 + 1;
241 is a prime and 241 = 30*8 + 1;
271 is a prime and 271 = 30*9 + 1.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A057204, A068228, A129805, A039949, A132231-A132236.

A132230 := Filtered(Filtered([1..10^6], n -> n mod 30 = 1), IsPrime); # Muniru A Asiru, Nov 01 2017

select(isprime, [seq(i,i=1..1000,30)]); # Robert Israel, Jan 19 2016

Select[Range[1, 3000, 30], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2012 *)
Select[Prime[Range[400]],Mod[#,30]==1&] (* Harvey P. Dale, May 21 2021 *)

is(n)=isprime(n) && n%30==1 \\ Charles R Greathouse IV, Jul 01 2016

a(n) = A111175(n)*30 + 1. - Ray Chandler, Apr 07 2009

Intersection of A030430 and A002476. - Ray Chandler, Apr 07 2009

Edited by Ray Chandler, Apr 07 2009

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

A001617 Genus of modular group Gamma_0(n). Or, genus of modular curve X_0(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 0, 2, 1, 2, 2, 3, 2, 1, 3, 3, 3, 1, 2, 4, 3, 3, 3, 5, 3, 4, 3, 5, 4, 3, 1, 2, 5, 5, 4, 4, 5, 5, 5, 6, 5, 7, 4, 7, 5, 3, 5, 9, 5, 7, 7, 9, 6, 5, 5, 8, 5, 8, 7, 11, 6, 7, 4, 9, 7, 11, 7, 10, 9, 9, 7, 11, 7, 10, 9, 11, 9, 9, 7, 7, 9, 7, 8, 15

Offset: 1

N. J. A. Sloane

Also the dimension of the space of cusp forms of weight two and level n. - Gene Ward Smith, May 23 2006

			G.f. = x^11 + x^14 + x^15 + x^17 + x^19 + x^20 + x^21 + 2*x^22 + 2*x^23 + ...

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gheorghe Coserea, Table of n, a(n) for n = 1..50000 (first 1000 terms from N. J. A. Sloane)
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.
Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
Steven R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]
Ralf Hemmecke, Peter Paule, and Silviu Radu, Construction of Modular Function Bases for Gamma_0(121) related to p*(11*n + 6), (2019).
Nicolas Allen Smoot, Computer algebra with the fifth operation: applications of modular functions to partition congruences, Ph. D. Thesis, Johannes Kepler Univ., Linz (Austria 2022), 33.
Index entries for sequences related to modular groups

Cf. A001615, A000089, A000086, A001616, A054728, A091401, A091403, A091404.

a := func< n | n lt 1 select 0 else Dimension( CuspForms( Gamma0(n), 2))>; /* Michael Somos, May 08 2015 */

nu2 := proc (n) # number of elliptic points of order two (A000089) local i, s; if modp(n,4) = 0 then RETURN(0) fi; s := 1; for i in divisors(n) do if isprime(i) and i > 2 then s := s*(1+eval(legendre(-1,i))) fi od; s end:
nu3 := proc (n) # number of elliptic points of order three (A000086) local d, s; if modp(n,9) = 0 then RETURN(0) fi; s := 1; for d in divisors(n) do if isprime(d) then s := s*(1+eval(legendre(-3,d))) fi od; s end:
nupara := proc (n) # number of parabolic cusps (A001616) local b, d; b := 0; for d to n do if modp(n,d) = 0 then b := b+eval(phi(gcd(d,n/d))) fi od; b end:
A001615 := proc(n) local i,j; j := n; for i in divisors(n) do if isprime(i) then j := j*(1+1/i); fi; od; j; end;
genx := proc (n) # genus of X0(n) (A001617) #1+1/12*psi(n)-1/4*nu2(n)-1/3*nu3(n)-1/2*nupara(n) end: 1+1/12*A001615(n)-1/4*nu2(n)-1/3*nu3(n)-1/2*nupara(n) end: # Gene Ward Smith, May 23 2006

nu2[n_] := Module[{i, s}, If[Mod[n, 4] == 0, Return[0]]; s = 1; Do[ If[ PrimeQ[i] && i > 2, s = s*(1 + JacobiSymbol[-1, i])], {i, Divisors[n]}]; s];
nu3[n_] := Module[{d, s}, If[Mod[n, 9] == 0, Return[0]]; s = 1; Do[ If[ PrimeQ[d], s = s*(1 + JacobiSymbol[-3, d])], {d, Divisors[n]}]; s];
nupara[n_] := Module[{b, d}, b = 0; For[d = 1, d <= n, d++, If[ Mod[n, d] == 0, b = b + EulerPhi[ GCD[d, n/d]]]]; b];
A001615[n_] := Module[{i, j}, j = n; Do[ If[ PrimeQ[i], j = j*(1 + 1/i)], {i, Divisors[n]}]; j];
genx[n_] := 1 + (1/12)*A001615[n] - (1/4)*nu2[n] - (1/3)*nu3[n] - (1/2)*nupara[n];
A001617 = Table[ genx[n], {n, 1, 102}] (* Jean-François Alcover, Jan 04 2012, after Gene Ward Smith's Maple program *)
a[ n_] := If[ n < 1, 0, 1 + Sum[ MoebiusMu[ d]^2 n/d / 12 - EulerPhi[ GCD[ d, n/d]] / 2, {d, Divisors @n}] - Count[(#^2 - # + 1)/n & /@ Range[n], ?IntegerQ]/3 - Count[ (#^2 + 1)/n & /@ Range[n], ?IntegerQ]/4]; (* Michael Somos, May 08 2015 *)

A000089(n) = {
  if (n%4 == 0 || n%4 == 3, return(0));
  if (n%2 == 0, n \= 2);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k,1] % 4 == 3, 0, 2));
};
A000086(n) = {
  if (n%9 == 0 || n%3 == 2, return(0));
  if (n%3 == 0, n \= 3);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k,1] % 3 == 2, 0, 2));
};
A001615(n) = {
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, (f[k,1]+1)),
     h = prod(k=1, fsz, f[k,1]));
  return((n*g)\h);
};
A001616(n) = {
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, f[k,1]^(f[k,2]\2) + f[k,1]^((f[k,2]-1)\2));
};
a(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
vector(102, n, a(n))  \\ Gheorghe Coserea, May 20 2016

a(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2.
From Gheorghe Coserea, May 20 2016: (Start)
limsup a(n) / (n*log(log(n))) = exp(Euler)/(2*Pi^2), where Euler is A001620.
a(n) >= (n-5*sqrt(n)-8)/12, with equality iff n = p^2 for prime p=1 (mod 12) (see A068228).
a(n) < n * exp(Euler)/(2*Pi^2) * (log(log(n)) + 2/log(log(n))) for n>=3 (see Csirik link).
(End)

A329963 Numbers k such that sigma(k) is not divisible by 3.

Original entry on oeis.org

1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 52, 57, 61, 63, 64, 67, 73, 75, 76, 79, 81, 84, 91, 93, 97, 100, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 144, 148, 151, 156, 157, 163, 171, 172, 175, 181, 183, 189, 192, 193, 199, 201, 208, 211, 217, 219, 223, 225, 228, 229

Offset: 1

John L. Drost, Nov 25 2019

nonn

A number k is in the sequence iff in its prime factorization, all primes p == 1 (mod 3) occur to such a power p^e that e != 2 (mod 3), and all primes == 2 (mod 3) occur to even powers. (3 can occur to any power.) This sequence is similar but not identical to many others; in particular, 343 is in this sequence, but not in A034022. (And here we don't have 196, although it is in A034022). - First sentence corrected and additional notes added by Antti Karttunen, Jul 03 2024, see also Robert Israel's Nov 09 2016 comment in A087943.

The asymptotic density of this sequence is 0 (Dressler, 1975). - Amiram Eldar, Jul 23 2020

Robert Israel, Table of n, a(n) for n = 1..10000
Tewodros Amdeberhan, Victor H. Moll, Vaishavi Sharma, and Diego Villamizar, Arithmetic properties of the sum of divisors, arXiv:2007.03088 [math.NT], 2020. See p. 15 ff. [Note: the "if and only if" condition given in the beginning of Theorem 7.1 is for A003136, not for this sequence. - _Antti Karttunen_, Jul 04 2024]
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Complement of A087943. Positions of zeros in A354100, nonzeros in A074941.
Cf. A000203, A353815 (characteristic function).
Setwise difference A003136 \ A088535.
Subsequences: A002476, A068228, A351537, A374135.
Cf. also A088232.
Not the same as A034022.

[k:k in [1..200]| DivisorSigma(1,k) mod 3 ne 0]; // Marius A. Burtea, Jan 02 2020

select(t -> numtheory:-sigma(t) mod 3 <> 0, [$1..200]); # Robert Israel, Jan 01 2020

Select[Range[200], !Divisible[DivisorSigma[1, #], 3] &] (* Amiram Eldar, Nov 25 2019 *)

isok(k) = (sigma(k) % 3) != 0; \\ Michel Marcus, Nov 26 2019

isA329963 = A353815; \\ Antti Karttunen, Jul 03 2024

More terms from Joshua Oliver, Nov 26 2019

Data section further extended up to a(71), to better differentiate from nearby sequences - Antti Karttunen, Jul 04 2024

A243655 Positive numbers that are primitively represented by the indefinite quadratic form x^2 - 3y^2 of discriminant 12.

Original entry on oeis.org

1, 6, 13, 22, 33, 37, 46, 61, 69, 73, 78, 94, 97, 109, 118, 121, 141, 142, 157, 166, 169, 177, 181, 193, 213, 214, 222, 229, 241, 249, 253, 262, 277, 286, 313, 321, 334, 337, 349, 358, 366, 373, 382, 393, 397, 409, 421, 429, 433, 438, 454, 457, 478, 481

Offset: 1

N. J. A. Sloane, Jun 11 2014

nonn

x^2+2xy-2y^2 is an equivalent form.

Will Jagy, C++ program Conway_Positive_All.cc to find all positive numbers represented by an indefinite binary quadratic form
Will Jagy, Sample output from Conway_Positive_All.cc
Will Jagy, C++ program Conway_Positive_Primitive.cc to find positive numbers primitively represented by an indefinite binary quadratic form
Will Jagy, Sample output from Conway_Positive_Prim.cc
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A084916 (all numbers represented), A068228.

Reap[For[n = 1, n < 500, n++, r = Reduce[x^2 - 3 y^2 == n, {x, y}, Integers]; If[r =!= False, If[AnyTrue[{x, y} /. {ToRules[r /. C[1] -> 0]}, CoprimeQ @@ # &], Print[n]; Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

A038874 Primes p such that 3 is a square mod p.

Original entry on oeis.org

2, 3, 11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167, 179, 181, 191, 193, 227, 229, 239, 241, 251, 263, 277, 311, 313, 337, 347, 349, 359, 373, 383, 397, 409, 419, 421, 431, 433, 443, 457, 467, 479, 491, 503, 541, 563, 577, 587, 599, 601

Offset: 1

N. J. A. Sloane

Also primes congruent to {1, 2, 3, 11} mod 12.
The subsequence p = 1 (mod 4) corresponds to A068228 and only these entries of a(n) are squares mod 3 (from the quadratic reciprocity law). - Lekraj Beedassy, Jul 21 2004
Largest prime factors of n^2 - 3. - Vladimir Joseph Stephan Orlovsky, Aug 12 2009
Aside from 2 and 3, primes p such that Legendre(3, p) = 1. Bolker asserts there are infinitely many of these primes. - Alonso del Arte, Nov 25 2015
The associated bases of the squares are 1, 0, 5, 4, 7, 15, 12, 11, 8, 28, 21, 13...: 1^2 = 3 -1*2, 0^2 = 3-1*3, 5^2 = 3+ 2*11, 4^2 = 3+1*13, 7^2 = 3+2*23, 15^2 = 3+6*37, 12^2 = 3+3*47,... - R. J. Mathar, Feb 23 2017

			11 is in the sequence since the equation x^2 - 11y = 3 has solutions, such as x = 5, y = 2.
13 is in the sequence since the equation x^2 - 13y = 3 has solutions, such as x = 4, y = 1.
17 is not in the sequence because x^2 - 17y = 3 has no solutions in integers; Legendre(3, 17) = -1.

Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 74, Theorem 25.3.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Tamara M. Lavshuk, Regular polygons and polyhedra over finite field, Mathematical Notes of NEFU, Vol 22 No 4 (2015). Mentions this sequence.

Cf. A002313, A033203, A038873, A045331, A057125.

If the first two terms are omitted we get A097933. A040101 is another sequence.

[p: p in PrimesUpTo(1200) | p mod 12 in [1, 2, 3, 11]]; // Vincenzo Librandi, Aug 08 2012

select(isprime, [2,3, seq(seq(6+s+12*i, s=[-5,5]),i=0..1000)]); # Robert Israel, Dec 23 2015

Select[Prime[Range[250]], MemberQ[{1, 2, 3, 11}, Mod[#, 12]] &] (* Vincenzo Librandi, Aug 08 2012 *)
Select[Flatten[Join[{2, 3}, Table[{12n - 1, 12n + 1}, {n, 50}]]], PrimeQ] (* Alonso del Arte, Nov 25 2015 *)

forprime(p=2, 1e3, if(issquare(Mod(3, p)), print1(p , ", "))) \\ Altug Alkan, Dec 04 2015

a(n) ~ 2n log n. - Charles R Greathouse IV, Nov 29 2016

More terms from Henry Bottomley, Aug 10 2000

cp's OEIS Frontend

A141123 Primes of the form -x^2+2*x*y+2*y^2 (as well as of the form 3*x^2+6*x*y+2*y^2).

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A068231 Primes congruent to 11 mod 12.

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A068229 Primes congruent to 7 (mod 12).

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A040117 Primes congruent to 5 (mod 12). Also primes p such that x^4 = 9 has no solution mod p.

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A132230 Primes congruent to 1 (mod 30).

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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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A141123 Primes of the form -x^2+2xy+2y^2 (as well as of the form 3x^2+6xy+2*y^2).

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.