A106856 -id:A106856 - cp's OEIS frontend

A139651 Primes of the form x^2 + 210*y^2.

211, 331, 379, 499, 571, 739, 1009, 1051, 1129, 1171, 1201, 1579, 1801, 2011, 2179, 2251, 2521, 2689, 2731, 2851, 3019, 3049, 3259, 3361, 3529, 3571, 3691, 3739, 3889, 3931, 4099, 4201, 4561, 4729, 5209, 5419, 5569, 5779, 5881, 6091, 6211

Offset: 1

T. D. Noe, Apr 29 2008

Discriminant=-840. See A139643 for more information.

The primes are congruent to {1, 121, 169, 211, 289, 331, 361, 379, 499, 529, 571, 739} (mod 840).

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(7000) | p mod 840 in {1, 121, 169, 211, 289, 331, 361, 379, 499, 529, 571, 739}]; // Vincenzo Librandi, Jul 28 2012

k:=210; [p: p in PrimesUpTo(7000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

QuadPrimes2[1, 0, 210, 10000] (* see A106856 *)

A139652 Primes of the form x^2 + 232*y^2.

Original entry on oeis.org

233, 241, 257, 281, 313, 353, 401, 457, 521, 593, 673, 761, 857, 929, 937, 953, 977, 1009, 1049, 1097, 1153, 1193, 1217, 1289, 1321, 1553, 1601, 1657, 1753, 1889, 1913, 2017, 2081, 2089, 2113, 2137, 2153, 2297, 2377, 2441, 2609, 2617, 2633

Offset: 1

T. D. Noe, Apr 29 2008

Discriminant=-928. See A139643 for more information.

The primes are congruent to {1, 9, 25, 33, 49, 57, 65, 81, 121, 129, 161, 169, 209, 225} (mod 232).

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 232 in {1, 9, 25, 33, 49, 57, 65, 81, 121, 129, 161, 169, 209, 225}]; // Vincenzo Librandi, Jul 28 2012

k:=232; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

QuadPrimes2[1, 0, 232, 10000] (* see A106856 *)

A139653 Primes of the form x^2 + 253*y^2.

Original entry on oeis.org

257, 269, 317, 353, 397, 449, 509, 577, 653, 829, 929, 1013, 1021, 1061, 1093, 1153, 1181, 1237, 1277, 1301, 1373, 1409, 1453, 1549, 1637, 1697, 1741, 1973, 2017, 2237, 2281, 2293, 2341, 2377, 2381, 2473, 2557, 2677, 2693, 2753, 2861, 2953

Offset: 1

T. D. Noe, Apr 29 2008

Discriminant=-1012. See A139643 for more information.

The primes are congruent to {1, 9, 25, 49, 81, 93, 133, 141, 169, 177, 185, 213, 225, 257, 265, 269, 289, 301, 317, 353, 357, 361, 377, 397, 441, 445, 449, 485, 489, 509, 533, 537, 553, 565, 577, 581, 625, 653, 669, 685, 729, 749, 785, 817, 829, 837, 841, 905, 929, 933, 949, 961, 969, 993, 1005} (mod 1012).

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)

k:=253; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

QuadPrimes2[1, 0, 253, 10000] (* see A106856 *)

A139665 Primes of the form x^2 + 840*y^2.

Original entry on oeis.org

1009, 1129, 1201, 1801, 2521, 2689, 3049, 3361, 3529, 3889, 4201, 4561, 4729, 5209, 5569, 5881, 6841, 7561, 7681, 8089, 8521, 8689, 8761, 8929, 9241, 9601, 9769, 10369, 12049, 12289, 12601, 12721, 12889, 13441, 13729, 14281, 14401, 14449, 15121, 15241

Offset: 1

T. D. Noe, Apr 29 2008

Discriminant = -3360. See A139643 for more information.
The primes are congruent to {1, 121, 169, 289, 361, 529} (mod 840).
Also, primes that in 1969 were unverified values for n for the Erdos-Straus conjecture (that 4/n = 1/x + 1/y + 1/z is always solvable in natural numbers), see Mordell 1969. - Ron Knott, Dec 11 2013
There are 273 terms < 100000 in this sequence. Of these, 59 are of the form 11n-3 or 11n-4. Since 4/(11*n-3)= 1/(3*n) + 1/(3*(11*n-3)) + 1/(n*(11*n-3)) and 4/(11*n-4)= 1/(3*n-1) + 1/(3*(11*n-4)) + 1/(3*(3*n-1)*(11*n-4)), these terms can be removed from the set of primes not proved for the Erdős-Straus conjecture. For example: 1009 = 11*92-3 so 4/1009 = 1/(3*92)+1/(3*1009)+1/(92*1009). These formulas were taken from the tables in Chapter 1 of the Mishima link. - Gary Detlefs, Jan 27 2014

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section D11.
L. J. Mordell, Diophantine Equations, Academic press, 1969, pages 287-290.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi).
Ron Knott, Egyptian Fractions.
H. H. Mishima, Overview of "Mathematician's Secret Room".
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).

Cf. A106856, A139643.

[ p: p in PrimesUpTo(15000) | p mod 840 in {1, 121, 169, 289, 361, 529}]; // Vincenzo Librandi, Jul 29 2012

k:=840; [p: p in PrimesUpTo(16000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

QuadPrimes2[1, 0, 840, 10000] (* see A106856 *)
Select[Table[Prime[n], {n, 1, 5000}], MemberQ[{1, 11^2, 13^2, 17^2, 19^2, 23^2}, Mod[#, 840]] &] (* Ron Knott, Dec 11 2013 *)

From Gary Detlefs, Jan 22 2014: (Start)
a(n) == {1,25,49,73} (mod 96);
a(n)^2 == {1,49} (mod 96);
a(n)^4 == 1 (mod 96). (End)
a(n) == {1,9,25} (mod 56). - Gary Detlefs, Jan 27 2014

A139831 Primes of the form 2x^2+2xy+23y^2.

Original entry on oeis.org

2, 23, 47, 83, 107, 167, 227, 263, 347, 383, 443, 467, 503, 563, 587, 647, 683, 743, 827, 863, 887, 947, 983, 1103, 1163, 1187, 1223, 1283, 1307, 1367, 1427, 1487, 1523, 1583, 1607, 1667, 1787, 1823, 1847, 1907, 2003, 2027, 2063, 2087, 2207

Offset: 1

T. D. Noe, May 02 2008

Discriminant=-180. See A139827 for more information.

Except for 2, also primes of the forms 3x^2+20y^2 (A107169) and 8x^2+4xy+23y^2. See A140633. - T. D. Noe, May 19 2008

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)

[2] cat[ p: p in PrimesUpTo(3000) | p mod 60 in {23, 47}]; // Vincenzo Librandi, Jul 29 2012

QuadPrimes2[2, -2, 23, 10000] (* see A106856 *)

Except for 2, the primes are congruent to {23, 47} (mod 60).

A139843 Primes of the form 6x^2 + 17y^2.

Original entry on oeis.org

17, 23, 41, 71, 113, 167, 233, 311, 401, 431, 449, 479, 503, 521, 617, 641, 719, 743, 809, 839, 857, 881, 887, 911, 929, 983, 1031, 1049, 1151, 1193, 1217, 1289, 1319, 1367, 1433, 1439, 1553, 1559, 1601, 1697, 1847, 2063, 2081, 2111, 2153, 2207

Offset: 1

T. D. Noe, May 02 2008

Discriminant = -408. See A139827 for more information.

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 408 in {17, 23, 41, 65, 71, 95, 113, 143, 167, 209, 215, 233, 311, 329, 335, 377, 401}]; // Vincenzo Librandi, Jul 29 2012

QuadPrimes2[6, 0, 17, 10000] (* see A106856 *)

list(lim)=my(v=List([17]), s=[23, 41, 65, 71, 95, 113, 143, 167, 209, 215, 233, 311, 329, 335, 377, 401]); forprime(p=23, lim, if(setsearch(s, p%408), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Feb 10 2017

The primes are congruent to {17, 23, 41, 65, 71, 95, 113, 143, 167, 209, 215, 233, 311, 329, 335, 377, 401} (mod 408).

A139850 Primes of the form 11x^2 + 8xy + 11y^2.

Original entry on oeis.org

11, 71, 179, 191, 239, 359, 431, 491, 599, 659, 911, 1019, 1031, 1439, 1451, 1499, 1619, 1871, 2039, 2111, 2339, 2459, 2531, 2591, 2699, 2711, 2879, 3011, 3119, 3299, 3371, 3539, 3719, 3851, 4019, 4139, 4211, 4271, 4391, 4691, 4799, 5051

Offset: 1

T. D. Noe, May 02 2008

Discriminant = -420. See A139827 for more information.

Also primes of the forms 11x^2 + 6xy + 39y^2 and 11x^2 + 10xy + 50y^2. See A140633. - T. D. Noe, May 19 2008

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(6000) | p mod 420 in {11, 71, 179, 191, 239, 359}]; // Vincenzo Librandi, Jul 29 2012

Union[QuadPrimes2[11, 8, 11, 10000], QuadPrimes2[11, -8, 11, 10000]] (* see A106856 *)

list(lim)=my(v=List(), s=[11, 71, 179, 191, 239, 359]); forprime(p=11, lim, if(setsearch(s, p%420), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Feb 10 2017

The primes are congruent to {11, 71, 179, 191, 239, 359} (mod 420).

A139851 Primes of the form 4x^2+4xy+29y^2.

Original entry on oeis.org

29, 37, 53, 109, 149, 197, 277, 317, 373, 389, 421, 541, 557, 613, 653, 701, 709, 757, 821, 877, 1061, 1093, 1117, 1213, 1229, 1373, 1381, 1429, 1453, 1493, 1549, 1597, 1621, 1709, 1733, 1789, 1877, 1901, 1933, 1997, 2053, 2069, 2213, 2221, 2237

Offset: 1

T. D. Noe, May 02 2008

Discriminant=-448. See A139827 for more information.

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 56 in {29, 37, 53}]; // Vincenzo Librandi, Jul 29 2012

QuadPrimes2[4, -4, 29, 10000] (* see A106856 *)

The primes are congruent to {29, 37, 53} (mod 56).

A139854 Primes of the form 3x^2 + 40y^2.

Original entry on oeis.org

3, 43, 67, 163, 283, 307, 523, 547, 643, 787, 883, 907, 1123, 1483, 1627, 1723, 1747, 1867, 1987, 2083, 2203, 2347, 2467, 2683, 2707, 2803, 3067, 3163, 3187, 3307, 3547, 3643, 3907, 4003, 4027, 4243, 4363, 4483, 4507, 4603, 4723, 4987, 5107

Offset: 1

T. D. Noe, May 02 2008

Discriminant=-480. See A139827 for more information.

Except for 3, also primes of the form 27x^2+12xy+28y^2. See A140633. - T. D. Noe, May 19 2008

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. A140633.

[3] cat [ p: p in PrimesUpTo(6000) | p mod 120 in {43, 67}]; // Vincenzo Librandi, Jul 29 2012

QuadPrimes2[3, 0, 40, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=1, sqrtint(lim\3), w=3*x^2; for(y=0, sqrtint((lim-w)\40), if(isprime(t=w+40*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 22 2017

Except for 3, the primes are congruent to {43, 67} (mod 120).

A139860 Primes of the form 12x^2+12xy+13y^2.

Original entry on oeis.org

13, 37, 157, 277, 373, 397, 613, 733, 757, 853, 877, 997, 1093, 1117, 1213, 1237, 1453, 1597, 1693, 1933, 2053, 2293, 2437, 2557, 2677, 2797, 2917, 3037, 3253, 3373, 3517, 3613, 3637, 3733, 3853, 3877, 4093, 4357, 4597, 4813, 4933, 4957

Offset: 1

T. D. Noe, May 02 2008

Discriminant=-480. See A139827 for more information.

Also primes of the forms 13x^2+2xy+37y^2 and 13x^2+4xy+28y^2. See A140633. - T. D. Noe, May 19 2008

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(6000) | p mod 120 in {13, 37}]; // Vincenzo Librandi, Jul 29 2012

QuadPrimes2[12, -12, 13, 10000] (* see A106856 *)

The primes are congruent to {13, 37} (mod 120).

Corrected and extended b-file - Ray Chandler, Jul 30 2014

cp's OEIS Frontend

A139651 Primes of the form x^2 + 210*y^2.

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A139652 Primes of the form x^2 + 232*y^2.

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A139653 Primes of the form x^2 + 253*y^2.

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A139665 Primes of the form x^2 + 840*y^2.

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A139831 Primes of the form 2x^2+2xy+23y^2.

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A139843 Primes of the form 6x^2 + 17y^2.

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A139850 Primes of the form 11x^2 + 8xy + 11y^2.

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A139851 Primes of the form 4x^2+4xy+29y^2.

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