A141168 -id:A141168 - cp's OEIS frontend

A141167 Primes of the form 8x^2+xy-8*y^2.

61, 67, 113, 157, 193, 197, 227, 241, 257, 419, 499, 587, 631, 643, 653, 739, 821, 823, 859, 863, 907, 929, 947, 971, 997, 1019, 1039, 1051, 1087, 1181, 1187, 1217, 1289, 1303, 1319, 1373, 1511, 1531, 1637, 1777, 1783, 1801, 1913, 1997, 2027, 2039, 2069, 2087, 2129, 2213

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jun 12 2008

nonn

Discriminant = 257. Class = 3. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.

			a(6)=197 because we can write 197 = 8*5^2+5*1-8*1^2.

Z. I. Borevich and I. R. Shafarevich, Number Theory

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
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.

Numbers of the form 8x^2+xy-8y^2 in A243180.
Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141111, A141112 (d=65). A141168 (d=257).
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

q := 8*x^2 + x*y - 8*y^2; pmax = 3000; xmax = xmax0 = 50; ymin = ymin0 = -50; ymax = ymax0 = 50; k = 1.3 (* expansion coeff. for maxima *) ; dx = dy = 2; prms0 = {}; prms = {2}; While[prms != prms0, xx = yy = {}; prms0 = prms; prms = Reap[Do[p = q; If[2 <= p <= pmax && PrimeQ[p], AppendTo[xx, x]; AppendTo[yy, y]; Sow[p]], {x, 1, If[xmax == xmax0, xmax, Floor[k*xmax]], dx}, {y, If[ymin == ymin0, ymin, Floor[k*ymin]], If[ymax == ymax0, ymax, Floor[k*ymax]]}, dy]][[2, 1]] // Union; xmax = Max[xx]; ymin = Min[yy]; ymax = Max[yy]; Print[Length[prms], " terms", "  xmax = ", xmax, "  ymin = ", ymin, "  ymax = ", ymax ]]; A141167 = prms (* Jean-François Alcover, Oct 26 2016 *)

# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([8, 1, -8])
print(Q.represented_positives(2213, 'prime')) # Peter Luschny, Oct 26 2016

A243180 Numbers of the form 8x^2+xy-8y^2.

Original entry on oeis.org

0, 1, 4, 8, 9, 16, 22, 25, 26, 32, 34, 36, 44, 46, 49, 52, 58, 61, 62, 64, 67, 68, 72, 81, 88, 92, 100, 104, 113, 116, 118, 121, 124, 128, 136, 143, 144, 146, 157, 158, 169, 176, 178, 184, 187, 193, 196, 197, 198, 200, 208, 221, 225, 227, 232, 234, 236, 241, 242, 244, 248, 253, 256, 257, 268, 272, 274, 278, 286, 288, 289, 292, 299, 306, 316, 319, 324, 338, 341

Offset: 1

N. J. A. Sloane, Jun 02 2014

nonn

Discriminant 257.

32*a(n) has the form z^2 - 257*y^2, where z = 16*x+y. [Bruno Berselli, Jun 20 2014]

R. J. Mathar, Table of n, a(n) for n = 1..1676
Peter Luschny, Binary Quadratic Forms
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Primes: A141167. Cf. A243181, A141168.

maxTerm = 400; m0 = 10; dm = 10; Clear[f]; f[m_] := f[m] = Table[8*x^2 + x*y - 8*y^2 , {x, -m, m}, {y, -m, m}] // Flatten // Union // Select[#, 0 <= # <= maxTerm&]&; f[m0]; f[m = m0]; While[f[m] != f[m - dm], m = m + dm]; f[m] (* Jean-François Alcover, Jun 04 2014 *)

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([8, 1, -8])
print([0]+Q.represented_positives(341)) # Peter Luschny, Oct 26 2016

A141750 Primes of the form 4x^2 + 3x*y - 4y^2 (as well as of the form 2x^2 + 9xy + y^2).

Original entry on oeis.org

2, 3, 19, 23, 37, 41, 61, 67, 71, 73, 79, 89, 97, 109, 127, 137, 149, 173, 181, 211, 223, 227, 251, 257, 269, 283, 293, 311, 317, 347, 349, 353, 359, 367, 373, 383, 389, 397, 401, 419, 439, 457, 461, 463, 479, 487, 499, 503, 509, 523, 547, 557, 587, 593, 607

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jul 03 2008

nonn

Discriminant = 73. Class = 1. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2-4ac.

Is this the same as A038957? - R. J. Mathar, Jul 04 2008. Answer: almost certainly - see the Tunnell notes in A033212. - N. J. A. Sloane, Oct 18 2014

			a(2) = 3 because we can write 3 = 4*1^2 + 3*1*1 - 4*1^2.

Z. I. Borevich and I. R. Shafarevich, Number Theory.

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.

See also A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141158 (d=20). A141159, A141160 (d=21). A141170, A141171 (d=24). A141172, A141173 (d=28). A141174, A141175 (d=32). A141176, A141177 (d=33). A141178 (d=37). A141179, A141180 (d=40). A141181 (d=41). A141182, A141183 (d=44). A033212, A141785 (d=45). A068228, A141187 (d=48). A141188 (d=52). A141189 (d=53). A141190, A141191 (d=56). A141192, A141193 (d=57). A107152, A141302, A141303, A141304 (d=60). A141215 (d=61). A141111, A141112 (d=65). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).

A141772 Primes of the form 3x^2 + 5xy - 5y^2 (as well as of the form 7x^2 + 13xy + 3y^2).

Original entry on oeis.org

3, 5, 7, 17, 23, 37, 73, 97, 107, 113, 163, 167, 173, 193, 197, 227, 233, 277, 283, 313, 317, 337, 347, 367, 397, 487, 503, 547, 607, 617, 643, 653, 673, 677, 683, 743, 787, 823, 827, 853, 857, 877, 887, 907, 947, 983, 997, 1013, 1093, 1117, 1153, 1163, 1187

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jul 04 2008

nonn

Discriminant = 85. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac.

			a(1) = 3 because we can write 3 = 3*1^2 + 5*1*0 - 5*0^2 (or 3 = 7*0^2 + 13*0*1 + 3*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.

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.

Cf. A141773 (d=85). See also A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141158 (d=20). A141159, A141160 (d=21). A141170, A141171 (d=24). A141172, A141173 (d=28). A141174, A141175 (d=32). A141176, A141177 (d=33). A141178 (d=37). A141179, A141180 (d=40). A141181 (d=41). A141182, A141183 (d=44). A033212, A141785 (d=45). A068228, A141187 (d=48). A141188 (d=52). A141189 (d=53). A141190, A141191 (d=56). A141192, A141193 (d=57). A107152, A141302, A141303, A141304 (d=60). A141215 (d=61). A141111, A141112 (d=65). A141750 (d=73). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).

More terms from Colin Barker, Apr 04 2015

Typo in crossrefs fixed by Colin Barker, Apr 05 2015

A243181 Numbers of the form 4x^2+9xy-11y^2.

Original entry on oeis.org

0, 2, 4, 8, 11, 13, 16, 17, 18, 22, 23, 26, 29, 31, 32, 34, 36, 44, 46, 50, 52, 58, 59, 62, 64, 68, 72, 73, 79, 88, 89, 92, 98, 99, 100, 104, 116, 117, 118, 121, 122, 124, 128, 134, 136, 137, 139, 143, 144, 146, 153, 158, 162, 169, 173, 176, 178, 184, 187, 196, 198, 199, 200, 207, 208, 211, 221, 223, 226, 232, 234, 236, 239, 242, 244, 248, 253, 256, 261, 268

Offset: 1

N. J. A. Sloane, Jun 02 2014

nonn

Discriminant 257.

16*a(n) has the form z^2 - 257*y^2, where z = 8*x+9*y. [Bruno Berselli, Jun 20 2014]

R. J. Mathar, Table of n, a(n) for n = 1..2180
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Primes: A141168.

maxTerm = 300; m0 = 10; dm = 10; Clear[f]; f[m_] := f[m] = Table[4*x^2 + 9 x*y - 11*y^2 , {x, -m, m}, {y, -m, m}] // Flatten // Union // Select[#, 0 <= # <= maxTerm&]&; f[m0]; f[m = m0]; While[f[m] != f[m - dm], m = m + dm]; f[m] (* Jean-François Alcover, Jun 04 2014 *) (* Brute force search, so not guaranteed to find all solutions, I believe. - N. J. A. Sloane, Jun 05 2014 *)
Reap[For[n = 0, n <= 30, n++,
   If[Reduce[4*x^2 + 9*x*y - 11*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]] (* Better program, not brute force, but slow. Confirms the terms up through 29. - N. J. A. Sloane, Jun 05 2014 *)

A141778 Primes of the form 4x^2 + 3x*y - 5y^2 (as well as of the form 8x^2 + 11xy + y^2).

Original entry on oeis.org

2, 5, 11, 17, 47, 53, 67, 71, 73, 79, 89, 97, 107, 109, 131, 139, 157, 167, 173, 179, 199, 223, 227, 233, 251, 257, 263, 269, 271, 277, 283, 307, 311, 317, 331, 347, 367, 373, 401, 409, 443, 449, 461, 463, 467, 479, 487, 509, 523, 587, 601, 607, 613, 619, 631

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jul 04 2008

nonn

Discriminant = 89. Class = 1. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1.

A subsequence of (and may possibly coincide with) A038977. - R. J. Mathar, Jul 22 2008

			a(1) = 2 because we can write 2 = 4*1^2 + 3*1*1 - 5*1^2.

Z. I. Borevich and I. R. Shafarevich, Number Theory.

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.

See also A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141158 (d=20). A141159, A141160 (d=21). A141170, A141171 (d=24). A141172, A141173 (d=28). A141174, A141175 (d=32). A141176, A141177 (d=33). A141178 (d=37). A141179, A141180 (d=40). A141181 (d=41). A141182, A141183 (d=44). A033212, A141785 (d=45). A068228, A141187 (d=48). A141188 (d=52). A141189 (d=53). A141190, A141191 (d=56). A141192, A141193 (d=57). A107152, A141302, A141303, A141304 (d=60). A141215 (d=61). A141111, A141112 (d=65). A141750 (d=73). A141772, A141773 (d=85). A141776, A141777 (d=88). A141778 (d=89). A141161, A141163 (d=148). A141165, A141166 (d=229). A141167, A141168 (d=257).

Typo in crossrefs fixed by Colin Barker, Apr 05 2015

cp's OEIS Frontend

A141167 Primes of the form 8*x^2+x*y-8*y^2.

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A243180 Numbers of the form 8x^2+xy-8y^2.

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A141750 Primes of the form 4*x^2 + 3*x*y - 4*y^2 (as well as of the form 2*x^2 + 9*x*y + y^2).

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A141772 Primes of the form 3*x^2 + 5*x*y - 5*y^2 (as well as of the form 7*x^2 + 13*x*y + 3*y^2).

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A243181 Numbers of the form 4x^2+9xy-11y^2.

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Mathematica

A141778 Primes of the form 4*x^2 + 3*x*y - 5*y^2 (as well as of the form 8*x^2 + 11*x*y + y^2).

Views

Author

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Comments

Examples

References

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Extensions

A141167 Primes of the form 8x^2+xy-8*y^2.

A141750 Primes of the form 4x^2 + 3x*y - 4y^2 (as well as of the form 2x^2 + 9xy + y^2).

A141772 Primes of the form 3x^2 + 5xy - 5y^2 (as well as of the form 7x^2 + 13xy + 3y^2).

A141778 Primes of the form 4x^2 + 3x*y - 5y^2 (as well as of the form 8x^2 + 11xy + y^2).