A106856 -id:A106856 - cp's OEIS frontend

A107218 Primes of the form 4x^2 + 25y^2.

29, 41, 61, 89, 229, 241, 281, 349, 421, 509, 601, 641, 661, 701, 709, 769, 809, 821, 881, 1009, 1049, 1109, 1181, 1201, 1229, 1249, 1289, 1301, 1321, 1381, 1409, 1481, 1549, 1669, 1709, 1789, 1801, 1901, 2029, 2069, 2089, 2141, 2161, 2221, 2281

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -400. See A107132 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)

QuadPrimes2[4, 0, 25, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=1, sqrtint(lim\4), w=4*x^2; for(y=1, sqrtint((lim-w)\25), if(isprime(t=w+25*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 10 2017

A107219 Primes of the form x^2 + 100y^2.

Original entry on oeis.org

101, 109, 149, 181, 269, 389, 401, 409, 449, 461, 521, 541, 569, 761, 829, 929, 941, 1021, 1061, 1069, 1129, 1361, 1429, 1489, 1601, 1609, 1621, 1721, 1741, 1861, 1889, 1949, 2081, 2129, 2269, 2309, 2441, 2549, 2609, 2621, 2689, 2749, 2789

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -400. See A107132 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)

QuadPrimes2[1, 0, 100, 10000] (* see A106856 *)

list(lim)=my(v=List(),w,t); for(x=1, sqrtint(lim\1), w=x^2; for(y=1, sqrtint((lim-w)\100), if(isprime(t=w+100*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 10 2017

A107628 Number of integral quadratic forms ax^2 + bxy + cy^2 whose discriminant b^2-4ac is -n, 0 <= b <= a <= c and gcd(a,b,c) = 1.

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 3, 2, 0, 0, 1, 2, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 3, 3, 0, 0, 2, 2, 0, 0, 3, 2, 0, 0, 1, 3, 0, 0, 4, 2, 0, 0, 2, 2, 0, 0, 3, 3, 0, 0, 2, 4, 0, 0, 4, 2, 0, 0, 2, 2, 0, 0, 5, 4, 0, 0, 2, 2, 0, 0, 3, 4, 0

Offset: 1

T. D. Noe, May 18 2005, Apr 30 2008

nonn

This sequence is closely related to the class number function, h(-n), which is given for fundamental discriminants in A006641. For a fundamental discriminant d, we have h(-d) < 2a(d). It appears that a(n) < Sqrt(n) for all n. For k>1, the primes p for which a(p)=k coincide with the numbers n such that the class number h(-n) is 2k-1 (see A006203, A046002, A046004, A046006. A046008, A046010, A046012, A046014, A046016 A046018, A046020). - T. D. Noe, May 07 2008

			a(15)=2 because the forms x^2 + xy + 4y^2 and 2x^2 + xy + 2y^2 have discriminant -15.

See A106856.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A106856 (start of many quadratic forms).
Cf. A133675 (n such that a(n)=1).
Cf. A223708 (without zeros).

dLim=150; cnt=Table[0, {dLim}]; nn=Ceiling[dLim/4]; Do[d=b^2-4a*c; If[GCD[a, b, c]==1 && 0<-d<=dLim, cnt[[ -d]]++ ], {b, 0, nn}, {a, b, nn}, {c, a, nn}]; cnt

{a(n)=local(m); if(n<3, 0, forvec(v=vector(3,k,[0,(n+1)\4]), if( (gcd(v)==1)&(-v[1]^2+4*v[2]*v[3]==n), m++ ), 1); m)} /* Michael Somos, May 31 2005 */

A139644 Primes of the form x^2 + 105*y^2.

Original entry on oeis.org

109, 421, 541, 709, 1009, 1129, 1201, 1381, 1429, 1549, 1621, 1789, 1801, 2221, 2269, 2389, 2521, 2689, 3049, 3061, 3109, 3229, 3301, 3361, 3469, 3529, 3889, 4201, 4561, 4621, 4729, 4789, 4909, 5209, 5569, 5581, 5749, 5821, 5881, 6301

Offset: 1

T. D. Noe, Apr 29 2008

Discriminant=-420. See A139643 for more information.

The primes are congruent to {1, 109, 121, 169, 289, 361} (mod 420).

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 420 in {1, 109, 121, 169, 289, 361}]; // Vincenzo Librandi, Jul 28 2012

k:=105; [p: p in PrimesUpTo(7000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

QuadPrimes2[1, 0, 105, 10000] (* see A106856 *)

A139645 Primes of the form x^2 + 112*y^2.

Original entry on oeis.org

113, 137, 193, 233, 281, 337, 401, 449, 457, 569, 617, 641, 673, 809, 953, 977, 1009, 1033, 1129, 1201, 1289, 1297, 1409, 1481, 1801, 1873, 1913, 2017, 2081, 2129, 2137, 2153, 2297, 2377, 2417, 2473, 2521, 2633, 2657, 2689, 2713, 2753, 2801

Offset: 1

T. D. Noe, Apr 29 2008

Discriminant is -448. See A139643 for more information.
Primes of the form 8*n + 1 which cannot be expressed as 7*k - 1, 7*k - 2, or 7*k - 4. a(n)^3 == 1 (mod 56). - Gary Detlefs, Jan 26 2014
The primes are congruent to {1, 9, 25} (mod 56).

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 {1, 9, 25}]; // Vincenzo Librandi, Jul 28 2012

k:=112; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

f:=n-> ceil((8*n+1)/7)-(8*n+1): for n from 1 to 350 do if isprime(8*n+1) and f(n)<>1 and f(n)<>2 and f(n)<>4 then print(8*n+1) fi od. # Gary Detlefs, Jan 26 2014

QuadPrimes2[1, 0, 112, 10000] (* see A106856 *)

A139646 Primes of the form x^2 + 130*y^2.

Original entry on oeis.org

131, 139, 179, 211, 251, 419, 491, 521, 569, 571, 601, 641, 659, 809, 859, 881, 971, 1049, 1091, 1171, 1249, 1291, 1361, 1459, 1481, 1499, 1531, 1609, 1699, 1811, 1889, 1979, 2011, 2081, 2089, 2129, 2131, 2161, 2339, 2441, 2521, 2531, 2539

Offset: 1

T. D. Noe, Apr 29 2008

Discriminant=-520. See A139643 for more information.

The primes are congruent to {1, 9, 49, 51, 81, 121, 129, 131, 139, 179, 209, 211, 251, 259, 289, 321, 329, 339, 361, 419, 441, 451, 459, 491} (mod 520).

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 520 in {1, 9, 49, 51, 81, 121, 129, 131, 139, 179, 209, 211, 251, 259, 289, 321, 329, 339, 361, 419, 441, 451, 459, 491}]; // Vincenzo Librandi, Jul 28 2012

k:=130; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

QuadPrimes2[1, 0, 130, 10000] (* see A106856 *)

A139647 Primes of the form x^2 + 133*y^2.

Original entry on oeis.org

137, 149, 197, 233, 277, 389, 457, 541, 557, 613, 617, 653, 701, 709, 757, 809, 821, 1033, 1061, 1201, 1213, 1289, 1297, 1373, 1429, 1453, 1493, 1597, 1621, 1733, 1873, 1901, 2053, 2069, 2129, 2137, 2153, 2213, 2221, 2297, 2381, 2417, 2437

Offset: 1

T. D. Noe, Apr 29 2008

Discriminant=-532. See A139643 for more information.

The primes are congruent to {1, 9, 25, 81, 85, 93, 121, 137, 149, 169, 177, 197, 225, 233, 253, 277, 289, 305, 309, 365, 389, 429, 457, 473, 501, 505, 529} (mod 532).

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 532 in {1, 9, 25, 81, 85, 93, 121, 137, 149, 169, 177, 197, 225, 233, 253, 277, 289, 305, 309, 365, 389, 429, 457, 473, 501, 505, 529}]; // Vincenzo Librandi, Jul 28 2012

k:=133; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

QuadPrimes2[1, 0, 133, 10000] (* see A106856 *)

A139648 Primes of the form x^2 + 165*y^2.

Original entry on oeis.org

181, 229, 421, 661, 709, 829, 1021, 1321, 1489, 1549, 1609, 1621, 1741, 2029, 2161, 2269, 2281, 2341, 2689, 3001, 3061, 3169, 3301, 3469, 3529, 4129, 4261, 4621, 4789, 4801, 4909, 5281, 5449, 5569, 5581, 5641, 5701, 6121, 6229, 6301, 6361

Offset: 1

T. D. Noe, Apr 29 2008

Discriminant = -660.

The primes are congruent to {1, 49, 169, 181, 229, 289, 301, 361, 421, 529} (mod 660).

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 660 in {1, 49, 169, 181, 229, 289, 301, 361, 421, 529}]; // Vincenzo Librandi, Jul 28 2012

k:=165; [p: p in PrimesUpTo(7000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

QuadPrimes2[1, 0, 165, 10000] (* see A106856 *)

A139649 Primes of the form x^2 + 177*y^2.

Original entry on oeis.org

181, 193, 241, 277, 373, 433, 577, 661, 709, 733, 757, 829, 853, 877, 997, 1069, 1201, 1237, 1549, 1597, 1609, 1621, 1657, 1669, 1693, 1777, 1789, 1933, 1993, 2113, 2269, 2293, 2377, 2389, 2557, 2617, 2677, 2749, 2833, 2857, 2917, 2953

Offset: 1

T. D. Noe, Apr 29 2008

Discriminant = -708.

The primes are congruent to {1, 25, 49, 85, 121, 133, 145, 169, 181, 193, 205, 241, 253, 265, 277, 289, 361, 373, 433, 481, 493, 517, 529, 553, 577, 625, 661, 685, 697} (mod 708).

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(4000) | p mod 708 in {1, 25, 49, 85, 121, 133, 145, 169, 181, 193, 205, 241, 253, 265, 277, 289, 361, 373, 433, 481, 493, 517, 529, 553, 577, 625, 661, 685, 697}]; // Vincenzo Librandi, Jul 28 2012

k:=177; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

QuadPrimes2[1, 0, 177, 10000] (* see A106856 *)

A139650 Primes of the form x^2 + 190*y^2.

Original entry on oeis.org

191, 199, 239, 271, 311, 359, 479, 631, 719, 761, 769, 809, 881, 919, 929, 1031, 1049, 1151, 1201, 1279, 1289, 1489, 1559, 1601, 1721, 1759, 1831, 1871, 1879, 1999, 2039, 2129, 2239, 2281, 2399, 2441, 2551, 2591, 2609, 2671, 2791, 2969

Offset: 1

T. D. Noe, Apr 29 2008

Discriminant = -760.

The primes are congruent to {1, 9, 39, 49, 81, 111, 119, 121, 159, 161, 169, 191, 199, 201, 239, 271, 289, 311, 321, 329, 351, 359, 391, 441, 479, 481, 511, 519, 529, 609, 631, 671, 681, 689, 719, 729} (mod 760).

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:=190; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

QuadPrimes2[1, 0, 190, 10000] (* see A106856 *)

cp's OEIS Frontend

A107218 Primes of the form 4x^2 + 25y^2.

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A107219 Primes of the form x^2 + 100y^2.

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A107628 Number of integral quadratic forms ax^2 + bxy + cy^2 whose discriminant b^2-4ac is -n, 0 <= b <= a <= c and gcd(a,b,c) = 1.

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A139644 Primes of the form x^2 + 105*y^2.

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Magma

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A139645 Primes of the form x^2 + 112*y^2.

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Magma

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A139646 Primes of the form x^2 + 130*y^2.

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Magma

Magma

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A139647 Primes of the form x^2 + 133*y^2.

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Magma

Magma

Mathematica

A139648 Primes of the form x^2 + 165*y^2.

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Magma