A139643 -id:A139643 - cp's OEIS frontend

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

13, 17, 29, 53, 61, 101, 113, 157, 173, 181, 233, 257, 269, 277, 313, 337, 373, 389, 433, 521, 569, 601, 641, 653, 673, 677, 701, 757, 797, 809, 829, 857, 881, 937, 953, 997, 1013, 1049, 1069, 1093, 1109, 1117

Offset: 1

N. J. A. Sloane

First differences are multiples of 4 (which follows from set of differences of the moduli in the Noe formula). Minimal difference 4 occurs at a(1)=17, a(25)=673, a(48)=1297, etc. - Zak Seidov, Oct 04 2014

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

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 3000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139643, A248212 (x) and A248213 (y).

QuadPrimes2[1, 0, 13, 10000] (* see A106856 *)

select(n->vecsearch([1,9,13,17,25,29,49],n%52), primes(100)) \\ Charles R Greathouse IV, Nov 09 2012

is_A033210(n)={vecsearch([1,9,13,17,25,29,49],n%52)&&isprime(n)} \\ setsearch() is still slower by a factor > 2. - M. F. Hasler, Oct 04 2014

Same as primes congruent to {1, 9, 13, 17, 25, 29, or 49} (mod 52). - T. D. Noe, Apr 29 2008 [See e.g. Cox, p. 36. - N. J. A. Sloane, May 27 2014]

a(n) ~ 4n log n. - Charles R Greathouse IV, Nov 09 2012

A033201 Primes of the form x^2 + 10*y^2.

Original entry on oeis.org

11, 19, 41, 59, 89, 131, 139, 179, 211, 241, 251, 281, 331, 379, 401, 409, 419, 449, 491, 499, 521, 569, 571, 601, 619, 641, 659, 691, 739, 761, 769, 809, 811, 859, 881, 929, 971, 1009, 1019, 1049, 1051, 1091, 1129, 1171, 1201, 1249, 1259, 1289, 1291, 1321, 1361, 1409, 1451, 1459, 1481

Offset: 1

N. J. A. Sloane

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 36.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 2000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139643.

Primes in A020673.

[p: p in PrimesUpTo(1500) | NormEquation(10,p) eq true]; // Bruno Berselli, Jul 03 2016

Clear[f,lst,p,x,y]; f[x_,y_]:=x^2+10*y^2; lst={};Do[Do[p=f[x,y];If[PrimeQ[p]&&p<7212,AppendTo[lst,p]],{y,0,6!}],{x,0,6!}];Take[Union[lst],222] (* Vladimir Joseph Stephan Orlovsky, Aug 04 2009 *)
QuadPrimes2[1, 0, 10, 10000] (* see A106856 *)

select(n->vecsearch([1,9,11,19],n%40), primes(100)) \\ Charles R Greathouse IV, Nov 09 2012

Same as primes congruent to 1, 9, 11, or 19 mod 40. See, e.g., Cox, p. 36.

a(n) ~ 4n log n. - Charles R Greathouse IV, Nov 09 2012

A033199 Primes of form x^2+6*y^2.

Original entry on oeis.org

7, 31, 73, 79, 97, 103, 127, 151, 193, 199, 223, 241, 271, 313, 337, 367, 409, 433, 439, 457, 463, 487, 577, 601, 607, 631, 673, 727, 751, 769, 823, 919, 937, 967, 991, 1009, 1033, 1039, 1063, 1087, 1129, 1153, 1201, 1231, 1249, 1279, 1297, 1303, 1321, 1327, 1399, 1423, 1447, 1471, 1489, 1543

Offset: 1

N. J. A. Sloane

Appears to also be the primes p such that p mod 6 = 1 and Fibonacci(p) mod 6 = 1. - Gary Detlefs, May 26 2014

N. J. A. Sloane and Vincenzo Librandi, Table of n, a(n) for n = 1..10000 (The first 2000 terms were found by Vincenzo Librandi)
David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989, p. 36.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139643, primes in A002481. Cf. A107006, A107008.

[p: p in PrimesUpTo(1600) | NormEquation(6,p) eq true]; // Bruno Berselli, Jul 03 2016

f[x_, y_] := x^2 + 6*y^2; lst = {}; Do[p = f[x, y]; If[ PrimeQ[ p], AppendTo[ lst, p]], {y, 20}, {x, 50}]; Take[ Union[ lst], 50] (* Vladimir Joseph Stephan Orlovsky, Aug 04 2009 *)

select(n->n%24==1||n%24==7, primes(100)) \\ Charles R Greathouse IV, Nov 09 2012

Same as primes congruent to 1 or 7 mod 24. See e.g. Cox, p. 36.

a(n) ~ 4n log n. - Charles R Greathouse IV, Nov 09 2012

Removed defective Mma program; extended the b-file using Charles R Greathouse's PARI program. - N. J. A. Sloane, Jun 06 2014

A033207 Primes of the form x^2 + 7*y^2.

Original entry on oeis.org

7, 11, 23, 29, 37, 43, 53, 67, 71, 79, 107, 109, 113, 127, 137, 149, 151, 163, 179, 191, 193, 197, 211, 233, 239, 263, 277, 281, 317, 331, 337, 347, 359, 373, 379, 389, 401, 421, 431, 443, 449, 457, 463, 487, 491

Offset: 1

N. J. A. Sloane

Except for a(1) = 7, these are the primes which can be written in the form a^2 + 7*b^2 with a > 0 and b > 0. - V. Raman, Sep 08 2012

These are the primes p for which p^3 - 1 is divisible by 7, with two exceptions: p = 2 is not in the sequence even though 2^3 - 1 is divisible by 7, and p = 7 is in the sequence even though 7^3 - 1 is not divisible by 7. Except for p = 7, if p^3 - 1 is not divisible by 7, it is congruent to 5 (mod 7). - Richard R. Forberg, Jun 03 2013

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

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Sushma Palimar and B. R. Shankar, Mersenne Primes in Real Quadratic Fields, Journal of Integer Sequences, Vol. 15 (2012), #12.5.6.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Essentially the same as A045373. Primes in A020670.

Cf. A002344, A002345, A139643.

QuadPrimes2[1, 0, 7, 10000] (* see A106856 *)

is(n)=kronecker(n,7)>=0 && isprime(n) && n>2 \\ Charles R Greathouse IV, Nov 19 2012

Primes congruent to {1, 7, 9, 11, 15, 23, 25} (mod 28). - T. D. Noe, Apr 29 2008

A106904 Primes of the form x^2-xy+13y^2, with x and y nonnegative.

Original entry on oeis.org

13, 19, 43, 67, 103, 127, 151, 157, 223, 229, 271, 307, 331, 349, 373, 409, 421, 433, 457, 463, 523, 577, 613, 631, 661, 727, 733, 739, 757, 769, 829, 859, 883, 919, 937, 967, 1021, 1033, 1039, 1063, 1069, 1087, 1123, 1171, 1237, 1249, 1279, 1291, 1327

Offset: 1

T. D. Noe, May 09 2005

Discriminant=-51.

Also: Primes which are squares (mod 51). Differs from the subsequence A106903 (because x^2+xy+y^2 = (x+y)^2 - (x+y)y + y^2) from a(20) = 463 on, A106903(20) = 523. Terms which are not in A106903 are: 463, 631, 1033, 1039, 1279, 1291,... Up to 1279 these are also in A139643. Cf. also A191034. - M. F. Hasler, Jan 15 2016

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, -1, 13, 10000] (* see A106856 *)

select(p->issquare(Mod(p,51))&&isprime(p),[1..1500]) \\ See A106903 for alternative code. - M. F. Hasler, Jan 15 2016

A139502 Primes of the form x^2 + 22x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

241, 409, 601, 769, 1009, 1129, 1201, 1249, 1321, 1489, 1609, 1801, 2089, 2161, 2281, 2521, 2689, 3001, 3049, 3121, 3169, 3361, 3529, 3769, 3889, 4129, 4201, 4441, 4561, 4729, 4801, 4969, 5209, 5281, 5449, 5521, 5569, 5641, 5689, 5881, 6121, 6361, 6481

Offset: 1

Artur Jasinski, Apr 24 2008

Also primes of the form x^2 + 120y^2. - T. D. Noe, Apr 29 2008
Also primes of the form x^2+240y^2. See A140633. - T. D. Noe, May 19 2008
In base 12, the sequence is 181, 2X1, 421, 541, 701, 7X1, 841, 881, 921, X41, E21, 1061, 1261, 1301, 13X1, 1561, 1681, 18X1, 1921, 1981, 1X01, 1E41, 2061, 2221, 2301, 2481, 2521, 26X1, 2781, 28X1, 2941, 2X61, 3021, 3081, 31X1, 3241, 3281, 3321, 3361, 34X1, 3661, 3821, 3901, where X is 10 and E is 11. Moreover, the discriminant is 340. - Walter Kehowski, Jun 01 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

Cf. A139643.

[ p: p in PrimesUpTo(7000) | p mod 120 in {1, 49}]; // Vincenzo Librandi, Jul 28 2012

QuadPrimes2[1, 0, 120, 10000] (* see A106856 *)

The primes are congruent to {1, 49} (mod 120). - T. D. Noe, Apr 29 2008

A033202 Primes of form x^2+93*y^2.

Original entry on oeis.org

97, 109, 157, 193, 349, 373, 397, 421, 541, 577, 661, 733, 769, 853, 877, 937, 997, 1033, 1093, 1117, 1213, 1237, 1249, 1321, 1489, 1597, 1609, 1621, 1657, 1693, 1741, 1777, 1861, 1993, 2017, 2029, 2053, 2113, 2221, 2281, 2341, 2389, 2437, 2521, 2593, 2713, 2797, 2857, 2953

Offset: 1

N. J. A. Sloane

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

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. A139643.

[p: p in PrimesUpTo(3000) | p mod 372 in {1,25,49,97, 109,121,133,157,169,193,205,253,289,349,361}]; // Vincenzo Librandi, Jul 02 2016

[p: p in PrimesUpTo(3000) | NormEquation(93,p) eq true]; // Bruno Berselli, Jul 03 2016

QuadPrimes2[1, 0, 93, 10000] (* see A106856 *)
Select[Prime@Range[500], MemberQ[{1, 25, 49, 97, 109, 121, 133, 157, 169, 193, 205, 253, 289, 349, 361}, Mod[#, 372]] &] (* Vincenzo Librandi, Jul 02 2016 *)

The primes are congruent to {1, 25, 49, 97, 109, 121, 133, 157, 169, 193, 205, 253, 289, 349, 361} (mod 372). - T. D. Noe, Apr 29 2008

A139506 Primes of the form x^2 + 26x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

193, 337, 457, 673, 1009, 1033, 1129, 1201, 1297, 1801, 1873, 2017, 2137, 2377, 2473, 2521, 2689, 2713, 2857, 3049, 3217, 3313, 3361, 3529, 3697, 3889, 4057, 4153, 4201, 4561, 4657, 4729, 4993, 5209, 5233, 5569, 5737, 5881, 6073, 6217, 6337, 6553, 6577

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Also primes of the form x^2 + 168y^2. - T. D. Noe, Apr 29 2008

In base 12, the sequence is 141, 241, 321, 481, 701, 721, 7X1, 841, 901, 1061, 1101, 1201, 12X1, 1461, 1521, 1561, 1681, 16X1, 17X1, 1921, 1X41, 1E01, 1E41, 2061, 2181, 2301, 2421, 24X1, 2521, 2781, 2841, 28X1, 2X81, 3021, 3041, 3281, 33X1, 34X1, 3621, 3721, 3801, 3961, 3981, where X is 10 and E is 11. Moreover, the discriminant is 480. - Walter Kehowski, Jun 01 2008

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

Cf. A139489, A007645, A068228, A007519, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

Cf. A139643.

a = {}; w = 26; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a]

The primes are congruent to {1, 25, 121} (mod 168). - T. D. Noe, Apr 29 2008

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 *)

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

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A033201 Primes of the form x^2 + 10*y^2.

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Magma

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A033199 Primes of form x^2+6*y^2.

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A033207 Primes of the form x^2 + 7*y^2.

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A106904 Primes of the form x^2-xy+13y^2, with x and y nonnegative.

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A139502 Primes of the form x^2 + 22x*y + y^2 for x and y nonnegative.

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Magma

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A033202 Primes of form x^2+93*y^2.

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Magma