A033210 -id:A033210 - 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

A248221 Numbers m such that 52*m + 1 is prime.

Original entry on oeis.org

1, 3, 6, 10, 13, 18, 21, 24, 25, 31, 36, 39, 40, 43, 45, 46, 49, 55, 60, 64, 66, 73, 78, 85, 91, 94, 96, 109, 115, 123, 124, 126, 129, 130, 133, 138, 139, 141, 144, 145, 151, 154, 159, 165, 168, 171, 174, 178, 181, 189, 193, 195, 196, 201, 211, 223, 225, 229

Offset: 1

Zak Seidov, Oct 04 2014

All terms are == {0,1} mod 3, because 52*(3k+2) + 1 is divisible by 3. - Zak Seidov, Oct 05 2014

Zak Seidov, Table of n, a(n) for n = 1..2000

Cf. A033210, A142508.

A248221:=n->`if`(isprime(52*n+1), n, NULL): seq(A248221(n), n=1..500); # Wesley Ivan Hurt, Oct 05 2014

Select[Range[250], PrimeQ[52# + 1] &] (* Alonso del Arte, Oct 04 2014 *)

for(n=1,10^3,if(isprime(52*n+1),print1(n,", "))) \\ Derek Orr, Oct 05 2014

a(n) = (A142508(n) - 1)/52.

More terms from Wesley Ivan Hurt, Oct 05 2014

A139642 Irregular triangle where row n gives the congruence (mod 4N) for the primes represented by the quadratic form x^2+Ny^2, where N=A000926(n) is a convenient number.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 7, 1, 5, 9, 13, 1, 5, 9, 1, 7, 1, 7, 9, 11, 15, 23, 25, 1, 9, 17, 25, 1, 13, 25, 1, 9, 11, 19, 1, 13, 25, 37, 1, 9, 13, 17, 25, 29, 49, 1, 19, 31, 49, 1, 9, 17, 25, 33, 41, 49, 57, 1, 19, 25, 43, 49, 67, 1, 25, 37, 1, 9, 15, 23, 25, 31, 47, 49, 71, 81, 1, 25, 49, 73, 1, 9

Offset: 1

T. D. Noe, Apr 28 2008

Each row begins with 1. For example, the 12th row is for N=13. The numbers in that row are 1, 9, 17, 25, 29 and 49, which means that the primes represented by the quadratic form x^2+13y^2 (A033210) are congruent to 1, 9, 17, 25, 29,or 49 (mod 52). Cox lists some of these congruences on page 36 of his book. As mentioned by Cox, for these N, every term of the congruence has the form b^2 or N+b^2 for some integer b. In some cases, the congruences can be simplified. For instance, for N=18 (A106950), the congruence is 1, 19, 25, 43, 49, 67 (mod 72), which can be simplified to 1, 19 (mod 24).

			1, 2,
1, 2, 3,
1, 3, 7,
1, 5, 9, 13,
1, 5, 9,
1, 7,
1, 7, 9, 11, 15, 23, 25,
1, 9, 17, 25,
1, 13, 25,
1, 9, 11, 19,
1, 13, 25, 37,
1, 9, 13, 17, 25, 29, 49,
1, 19, 31, 49,
1, 9, 17, 25, 33, 41, 49, 57,
1, 19, 25, 43, 49, 67,
1, 25, 37,
1, 9, 15, 23, 25, 31, 47, 49, 71, 81,
1, 25, 49, 73,
...

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, Section 3.

T. D. Noe, Rows n=1..65 of triangle, flattened
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

See the Binary Quadratic Forms and OEIS link for full list of primes generated by x^2+Ny^2, where N is a convenient number.

A247881 Numbers of the form x^2 + 13*y^2.

Original entry on oeis.org

0, 1, 4, 9, 13, 14, 16, 17, 22, 25, 29, 36, 38, 49, 52, 53, 56, 61, 62, 64, 68, 77, 81, 88, 94, 100, 101, 113, 116, 117, 118, 121, 126, 133, 134, 142, 144, 152, 153, 157, 166, 169, 173, 181, 182, 196, 198, 208, 209, 212, 217, 221, 224, 225, 233, 238, 244, 248, 256, 257, 261, 269

Offset: 1

Alonso del Arte, Sep 25 2014

Norms of numbers in Z[sqrt(-13)].

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A033210 (subsequence of primes).

N:= 1000: # for terms <= N
S:= {seq(seq(x^2 + 13*y^2, x=0 .. floor(sqrt(N-13*y^2))),y=0..floor(sqrt(N/13)))}:
sort(convert(S,list)); # Robert Israel, Jun 19 2025

A248019 Values of x in equation A142508(n)=x^2+13y^2.

Original entry on oeis.org

1, 12, 14, 14, 25, 27, 25, 14, 1, 40, 1, 27, 38, 40, 12, 14, 1, 53, 14, 1, 51, 40, 27, 64, 64, 14, 66, 64, 77, 77, 79, 66, 79, 77, 25, 38, 77, 40, 1, 79, 64, 53, 12, 92, 90, 51, 66, 77, 25, 64, 92, 77, 1, 79, 64, 53, 1, 103, 38, 12, 14, 53, 1, 77, 116, 79, 116, 92, 118, 118, 77, 103, 66, 118, 38

Offset: 1

Zak Seidov, Oct 06 2014

nonn

			a(1)=1 because A142508(1)=53=1^2+13*2^2 (x=1,y=2);
a(2)=12 because A142508(2)=157=12^2+13*1^2 (x=12, y=1).

Cf. A033210, A142508, A248221, A248019(values of y).

f[n_] := FindInstance[n == x^2 + 13 y^2 && x > 0 && y > 0, {x, y}, Integers][[1, 1, 2]]; f@# & /@ Select[ Prime@ Range@ 1840, Mod[#, 52] == 1 &] (* Robert G. Wilson v, Oct 06 2014 *)

A248368 Primes p such that 52*p + 1 is prime.

Original entry on oeis.org

3, 13, 31, 43, 73, 109, 139, 151, 181, 193, 211, 223, 229, 283, 349, 379, 409, 421, 463, 523, 601, 619, 691, 769, 823, 853, 1021, 1033, 1069, 1153, 1231, 1279, 1303, 1453, 1459, 1471, 1531, 1663, 1693, 1723, 1741, 1783, 1831, 1873, 1933, 2029, 2131, 2251, 2269, 2293, 2593, 2671, 2749, 2791

Offset: 1

Zak Seidov, Oct 05 2014

Or, primes in A248221. Subsequence of A248221. Note that a(1..6) coincide with A171517(1..6).

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A033210, A142508, A171517, A248221.

A248368:=n->`if`(isprime(52*n+1) and isprime(n), n, NULL): seq(A248368(n), n=1..4000); # Wesley Ivan Hurt, Oct 05 2014

s = {}; Do[If[PrimeQ[1 + 52*(p = Prime[n])], AppendTo[s, p]], {n, 500}]; s
Select[Prime[Range[500]],PrimeQ[52#+1]&] (* Harvey P. Dale, Aug 15 2017 *)

forprime(p=1,10^4,if(isprime(52*p+1),print1(p,", "))) \\ Derek Orr, Oct 05 2014

A248372 Numbers m such that both p = 52m + 1 and q = 52p + 1 are prime.

Original entry on oeis.org

36, 39, 60, 126, 171, 189, 195, 300, 315, 405, 420, 435, 504, 540, 570, 606, 720, 756, 816, 876, 960, 1089, 1221, 1224, 1260, 1329, 1365, 1371, 1389, 1404, 1530, 1554, 1674, 1740, 1785, 1791, 1914, 1959, 2085, 2244, 2304, 2334, 2376, 2451, 2454, 2520, 2631, 2646, 2715, 2799, 2976

Offset: 1

Zak Seidov, Oct 05 2014

nonn

All terms are divisible by 3, because if m == 1 or 2 (mod 3), either q or p is divisible by 3.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Subsequence of A248221.

Cf. A033210, A142508, A171517, A248221, A248368.

s={};Do[If[PrimeQ[p=52*n+1]&&PrimeQ[52*p+1],AppendTo[s,n]],{n,3000}];s
Select[Range[3000],AllTrue[{52#+1,53+2704#},PrimeQ]&] (* Harvey P. Dale, Mar 21 2025 *)

for(n=1,10^4,p=52*n+1;if(isprime(p)&&isprime(52*p+1),print1(n,", "))) \\ Derek Orr, Oct 06 2014

A248409 Least prime of the form x^2+13*n^2.

Original entry on oeis.org

13, 53, 181, 233, 389, 757, 641, 857, 1069, 1301, 1609, 1873, 2213, 2549, 3121, 3329, 3761, 4261, 4729, 5209, 5737, 6301, 6977, 7489, 8161, 8837, 9733, 10193, 10937, 11701, 12497, 13313, 14173, 15053, 16069, 17137, 18121, 18773, 19777, 20809, 21997, 23053, 24137, 25169, 27109, 27509

Offset: 1

Zak Seidov, Oct 06 2014

nonn

Subsequence of A033210 (Primes of the form x^2+13*y^2).

			a(1)=13 because the least prime of the form x^2+13*1^2 is 13 (at x=0).
a(100)=130121 because the least prime of the form x^2+13*100^2 is 130121 (at x=11).
a(200)=520361 because the least prime of the form x^2+13*200^2 is 520361 (at x=19).

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A033210, A106856, A247881, A248221.

a(n) = {x = 0; while (!isprime(p=x^2+13*n^2), x++); p;} \\ Michel Marcus, Oct 06 2014

Showing 1-10 of 12 results. Next

cp's OEIS Frontend

A248213 Values of y in A033210 (primes of the form x^2+13*y^2).

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A248212 Values of x in A033210 (primes of the form x^2+13*y^2).

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

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A248221 Numbers m such that 52*m + 1 is prime.

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A139642 Irregular triangle where row n gives the congruence (mod 4N) for the primes represented by the quadratic form x^2+Ny^2, where N=A000926(n) is a convenient number.

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A247881 Numbers of the form x^2 + 13*y^2.

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A248019 Values of x in equation A142508(n)=x^2+13y^2.

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A248368 Primes p such that 52*p + 1 is prime.

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A248372 Numbers m such that both p = 52*m + 1 and q = 52*p + 1 are prime.

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A248372 Numbers m such that both p = 52m + 1 and q = 52p + 1 are prime.