A033201 -id:A033201 - 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

A020673 Numbers of form x^2 + 10 y^2.

Original entry on oeis.org

0, 1, 4, 9, 10, 11, 14, 16, 19, 25, 26, 35, 36, 40, 41, 44, 46, 49, 56, 59, 64, 65, 74, 76, 81, 89, 90, 91, 94, 99, 100, 104, 106, 110, 115, 121, 126, 131, 139, 140, 144, 154, 160, 161, 164, 169, 171, 176, 179, 184, 185, 190, 196, 206, 209, 211, 224, 225, 234, 235, 236, 241

Offset: 1

David W. Wilson

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

Primes: A033201.

A216579 Number of positive integer solutions to the equation a^2 + 10*b^2 = n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2

Offset: 1

V. Raman, Sep 08 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A020673, A033201.

a[ n_] := If[ n < 1, 0, Length@Select[ Range@Sqrt[ n/10], IntegerQ@Sqrt[ n - 10 #] &]]; (* Michael Somos, Jan 10 2015 *)

a(n)=sum(b=1,sqrtint((n-1)\10),issquare(n-10*b^2)) \\ Charles R Greathouse IV, Nov 19 2014

A139829 Primes of the form 4x^2+4xy+11y^2.

Original entry on oeis.org

11, 19, 59, 131, 139, 179, 211, 251, 331, 379, 419, 491, 499, 571, 619, 659, 691, 739, 811, 859, 971, 1019, 1051, 1091, 1171, 1259, 1291, 1451, 1459, 1499, 1531, 1571, 1579, 1619, 1699, 1811, 1931, 1979, 2011, 2099, 2131, 2179, 2251, 2339, 2371

Offset: 1

T. D. Noe, May 02 2008

Discriminant=-160. See A139827 for more information.

Also, primes of form u^2+10v^2 with odd v, while A107145 has even v. One can transform its form as (2x+y)^2+10y^2 (where y can only be odd) and the latter is x^2+10(2y)^2. This sequence has primes {11,19} mod 20 while the second has {1,9} mod 20 and together they are the primes x^2+10y^2 (A033201) which are {1,9,11,20} mod 20. [From Tito Piezas III, Jan 01 2009]

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)

[ p: p in PrimesUpTo(3000) | p mod 40 in {11, 19}]; // Vincenzo Librandi, Jul 29 2012

QuadPrimes2[4, -4, 11, 10000] (* see A106856 *)

The primes are congruent to {11, 19} (mod 40).

A216577 Number of nonnegative integer solutions to the equation x^2 + 10*y^2 = n.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 1, 0, 0, 0, 0, 1

Offset: 0

V. Raman, Sep 08 2012

nonn

Cf. A020673, A033201.

Corrected by T. D. Noe and N. J. A. Sloane, Sep 10 2012

A317641 Expansion of theta_3(q)*theta_3(q^10), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 4, 0, 0, 4, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 2, 4, 0, 0, 4, 0, 4, 0, 0, 6, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 2, 8, 0, 0, 4, 0, 0, 0, 0, 4, 2

Offset: 0

Ilya Gutkovskiy, Aug 02 2018

nonn

Number of integer solutions to the equation x^2 + 10*y^2 = n.

			G.f. = 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^10 + 4*q^11 + 4*q^14 + 2*q^16 + 4*q^19 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A000024, A020673, A033201, A033723, A216577, A216579, A258034.

nmax = 100; CoefficientList[Series[EllipticTheta[3, 0, q] EllipticTheta[3, 0, q^10], {q, 0, nmax}], q]
nmax = 100; CoefficientList[Series[QPochhammer[-q, -q] QPochhammer[-q^10, -q^10]/(QPochhammer[q, -q] QPochhammer[q^10, -q^10]), {q, 0, nmax}], q]

G.f.: Product_{k>=1} (1 + x^(2*k-1))^2*(1 - x^(2*k))*(1 + x^(20*k-10))^2*(1 - x^(20*k)).

A216578 Prime numbers which cannot be written in the form x^2 + 10*y^2.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 29, 31, 37, 43, 47, 53, 61, 67, 71, 73, 79, 83, 97, 101, 103, 107, 109, 113, 127, 137, 149, 151, 157, 163, 167, 173, 181, 191, 193, 197, 199, 223, 227, 229, 233, 239, 257, 263, 269, 271, 277, 283, 293, 307, 311, 313, 317, 337, 347

Offset: 1

V. Raman, Sep 08 2012

nonn

Prime numbers not congruent to {1, 9, 11, 19} mod 40.

Cf. A020673, A033201.

A216580 Numbers which can be written in the form x^2 + 10*y^2, where x > 0 and y > 0.

Original entry on oeis.org

11, 14, 19, 26, 35, 41, 44, 46, 49, 56, 59, 65, 74, 76, 89, 91, 94, 99, 104, 106, 110, 115, 121, 126, 131, 139, 140, 154, 161, 164, 169, 171, 176, 179, 184, 185, 190, 196, 206, 209, 211, 224, 234, 235, 236, 241, 251, 254, 259, 260, 265, 266, 275, 281, 286

Offset: 1

V. Raman, Sep 08 2012

nonn

Cf. A020673, A033201.

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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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A020673 Numbers of form x^2 + 10 y^2.

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A216579 Number of positive integer solutions to the equation a^2 + 10*b^2 = n.

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A139829 Primes of the form 4x^2+4xy+11y^2.

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A216577 Number of nonnegative integer solutions to the equation x^2 + 10*y^2 = n.

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A317641 Expansion of theta_3(q)*theta_3(q^10), where theta_3() is the Jacobi theta function.

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Mathematica

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A216578 Prime numbers which cannot be written in the form x^2 + 10*y^2.

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A216580 Numbers which can be written in the form x^2 + 10*y^2, where x > 0 and y > 0.

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