A141111 -id:A141111 - cp's OEIS frontend

A068228 Primes congruent to 1 (mod 12).

13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349, 373, 397, 409, 421, 433, 457, 541, 577, 601, 613, 661, 673, 709, 733, 757, 769, 829, 853, 877, 937, 997, 1009, 1021, 1033, 1069, 1093, 1117, 1129, 1153, 1201, 1213, 1237, 1249, 1297

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

This has several equivalent definitions (cf. the Tunnell link)
Also primes of the form x^2 + 9y^2 (discriminant -36). - T. D. Noe, May 07 2005 [corrected by Klaus Purath, Jan 18 2023]
Also primes of the form x^2 - 12y^2 (discriminant 48). Cf. A140633. - T. D. Noe, May 19 2008 [corrected by Klaus Purath, Jan 18 2023]
Also primes of the form x^2 + 4*x*y + y^2.
Also primes of the form x^2 + 2*x*y - 2*y^2 (cf. A084916).
Also primes of the form x^2 + 6*x*y - 3*y^2.
Also primes of the form 4*x^2 + 8*x*y + y^2.
Also primes of the form u^2 - 3v^2 (use the transformation {u,v} = {x+2y,y}). - Tito Piezas III, Dec 28 2008
Sequence lists generalized cuban primes (A007645) that are the sum of 2 nonzero squares. - Altug Alkan, Nov 25 2015
Yasutoshi Kohmoto observes that prevprime(a(n)) is more frequently congruent to 3 (mod 4) than to 1. This bias can be explained by the possible prime constellations and gaps: To have the same residue mod 4 as a prime in the list, the previous prime must be at a gap of 4 or 8 or 12 ..., but a gap of 4 is impossible because 12k + 1 - 4 is divisible by 3, and gaps >= 12 are very rare for small primes. To have the residue 3 (mod 4) the previous prime can be at a gap of 2 or 6 with no a priori divisibility property. However, this bias tends to disappear as the primes (and average prime gaps) grow bigger: for primes < 10^5, the ratio is about 35% vs. 65% as the above simple explanation suggests, but considering primes up to 10^8 yields a ratio of about 41% vs. 59%. It can be expected that the ratio asymptotically tends to 1:1. - M. F. Hasler, Sep 01 2017
Also primes of the form x^2 - 27*y^2. - Klaus Purath, Jan 18 2023

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy, with permission]
Michael Penn, an example right from my number theory class., YouTube video, 2021.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
J. B. Tunnell, Proofs of Conjectures Concerning Entry A033212, the Sequence of Primes Congruent to 1 or 19 (mod 30)
J. Voight, Quadratic forms that represent almost the same primes, Math. Comp., Vol. 76 (2007), pp. 1589-1617.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A068227, A068229, A040117, A068231, A068232, A068233, A068234, A068235, A139643, A141122, A140633, A264732.
Subsequence of A084916.
Subsequence of A007645.
Also primes in A084916, A020672.
Cf. A141123 (d=12), A141111, A141112 (d=65), A141187 (d=48) A038872 (d=5), A038873 (d=8), A038883 (d=13), A038889 (d=17).
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

[p: p in PrimesUpTo(1400) | p mod 12 in {1}]; // Vincenzo Librandi, Jul 14 2012
For other programs see the "Binary Quadratic Forms and OEIS" link.

select(isprime, [seq(i,i=1..10000, 12)]); # Robert Israel, Nov 27 2015

Select[Prime/@Range[250], Mod[ #, 12]==1&]
Select[Range[13, 10^4, 12], PrimeQ] (* Zak Seidov, Mar 21 2011 *)

for(i=1,250, if(prime(i)%12==1, print(prime(i))))

forstep(p=13,10^4,12,isprime(p)&print(p)); \\ Zak Seidov, Mar 21 2011

Edited by Dean Hickerson, Feb 27 2002

Entry revised by N. J. A. Sloane, Oct 18 2014 (Edited, merged with A141122, submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 05 2008).

A007519 Primes of form 8n+1, that is, primes congruent to 1 mod 8.

Original entry on oeis.org

17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353, 401, 409, 433, 449, 457, 521, 569, 577, 593, 601, 617, 641, 673, 761, 769, 809, 857, 881, 929, 937, 953, 977, 1009, 1033, 1049, 1097, 1129, 1153, 1193, 1201, 1217, 1249, 1289, 1297, 1321, 1361

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Discriminant is 32, class is 2. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.
Integers n (n > 9) of form 4k + 1 such that binomial(n-1, (n-1)/4) == 1 (mod n) - Benoit Cloitre, Feb 07 2004
Primes of the form x^2 + 8y^2. - T. D. Noe, May 07 2005
Also primes of the form x^2 + 16y^2. See A140633. - T. D. Noe, May 19 2008
Is this the same sequence as A141174?
Being a subset of A001132 and also a subset of A038873, this is also a subset of the primes of the form u^2 - 2v^2. - Tito Piezas III, Dec 28 2008
These primes p are only which possess the property: for every integer m from interval [0, p) with the Hamming distance D(m, p) = 2, there exists an integer h from (m, p) with D(m, h) = 2. - Vladimir Shevelev, Apr 18 2012
Primes p such that p XOR 6 = p + 6. - Brad Clardy, Jul 22 2012
Odd primes p such that -1 is a 4th power mod p. - Eric M. Schmidt, Mar 27 2014
There are infinitely many primes of this form. See Brubaker link. - Alonso del Arte, Jan 12 2017
These primes split in Z[sqrt(2)]. For example, 17 = (-1)(1 - 3*sqrt(2))(1 + 3*sqrt(2)). This is also true of primes of the form 8n - 1. - Alonso del Arte, Jan 26 2017

			a(1) = 17 = 2 * 8 + 1 = (10001)_2. All numbers m from [0, 17) with the Hamming distance D(m, 17) = 2 are 0, 3, 5, 9. For m = 0, we can take h = 3, since 3 is drawn from (0, 17) and D(0, 3) = 2; for m = 3, we can take h = 5, since 5 from (3, 17) and D(3, 5) = 2; for m = 5, we can take h = 6, since 6 from (5, 17) and D(5, 6) = 2; for m = 9, we can take h = 10, since 10 is drawn from (9, 17) and D(9, 10) = 2. - _Vladimir Shevelev_, Apr 18 2012

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
Z. I. Borevich and I. R. Shafarevich, Number Theory.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 261.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Ben Brubaker, 18.781, Fall 2007 Problem Set 5: Solutions to Selected Problems, MIT (2007).
Peter Luschny, Binary Quadratic Forms
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 521.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Subsequence of A017077 and of A038873.
Cf. A139643. Complement in primes of A154264. Cf. A042987.
Cf. A065091, A002144, A094407, A133870, A142925, A208177, A208178, A076339.
Cf. A038872 (d = 5). A038873 (d = 8). A068228, A141123 (d = 12). A038883 (d = 13). A038889 (d = 17). A141111, A141112 (d = 65).
Cf. also A242663.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

a007519 n = a007519_list !! (n-1)
a007519_list = filter ((== 1) . a010051) [1,9..]
-- Reinhard Zumkeller, Mar 06 2012

[p: p in PrimesUpTo(2000) | p mod 8 eq 1 ]; // Vincenzo Librandi, Aug 21 2012

Select[1 + 8 Range@ 170, PrimeQ] (* Robert G. Wilson v *)

forprime(p=2,1e4,if(p%8==1,print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011

forprimestep(p=17,10^4,8, print1(p", ")) \\ Charles R Greathouse IV, Jul 17 2024

lista(nn)= my(vpr = []); for (x = 0, nn, y = 0; while ((v = x^2+6*x*y+y^2) < nn, if (isprime(v), if (! vecsearch(vpr, v), vpr = concat(vpr, v); vpr = vecsort(vpr););); y++;);); vpr; \\ Michel Marcus, Feb 01 2014

A007519_upto(N, start=1)=select(t->t%8==1,primes([start,N]))
#A7519=A007519_upto(10^5)
A007519(n)={while(#A7519A007519_upto(N*3\2, N+1))); A7519[n]} \\ M. F. Hasler, May 22 2025

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 4, -4])
print(Q.represented_positives(1361, 'prime'))  # Peter Luschny, Jan 26 2017

A007522 Primes of the form 8n+7, that is, primes congruent to -1 mod 8.

Original entry on oeis.org

7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263, 271, 311, 359, 367, 383, 431, 439, 463, 479, 487, 503, 599, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 887, 911, 919, 967, 983, 991, 1031, 1039, 1063, 1087, 1103, 1151

Offset: 1

N. J. A. Sloane and Robert G. Wilson v

Primes that are the sum of no fewer than four positive squares.
Discriminant is 32, class is 2. Binary quadratic forms ax^2 + bxy + cy^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.
Primes p such that x^4 = 2 has just two solutions mod p. Subsequence of A040098. Solutions mod p are represented by integers from 0 to p - 1. For p > 2, i is a solution mod p of x^4 = 2 if and only if p - i is a solution mod p of x^4 = 2, so the sum of the two solutions is p. The solutions are given in A065907 and A065908. - Klaus Brockhaus, Nov 28 2001
As this is a subset of A001132, this is also a subset of the primes of form x^2 - 2y^2. And as this is also a subset of A038873, this is also a subset of the primes of form x^2 - 2y^2. - Tito Piezas III, Dec 28 2008
Subsequence of A141164. - Reinhard Zumkeller, Mar 26 2011
Also a subsequence of primes of the form x^2 + y^2 + z^2 + 1. - Arkadiusz Wesolowski, Apr 05 2012
Primes p such that p XOR 6 = p - 6. - Brad Clardy, Jul 22 2012

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Subsequence of A004771.
Cf. A040098, A014754, A065907, A065908, A010051.
Cf. A141174 (d = 32). A038872 (d = 5). A038873 (d = 8). A068228, A141123 (d = 12). A038883 (d = 13). A038889 (d = 17). A141111, A141112 (d = 65).

a007522 n = a007522_list !! (n-1)
a007522_list = filter ((== 1) . a010051) a004771_list
-- Reinhard Zumkeller, Jan 29 2013

[p: p in PrimesUpTo(2000) | p mod 8 eq 7]; // Vincenzo Librandi, Jun 26 2014

select(isprime, [seq(i,i=7..10000,8)]); # Robert Israel, Nov 22 2016

Select[8Range[200] - 1, PrimeQ] (* Alonso del Arte, Nov 07 2016 *)

(A007522(m) = local(p, s, x, z); forprime(p = 3, m, s = []; for(x = 0, p-1, if(x^4%p == 2%p, s = concat(s, [x]))); z = matsize(s)[2]; if(z == 2, print1(p, ", ")))); A007522(1400)  \\ Does not return a(m) but prints all terms <= m. - Edited to make it executable by M. F. Hasler, May 22 2025.

A007522_upto(N, start=1)=select(p->p%8==7, primes([start, N]))
#A7522=A007522_upto(10^5)
A007522(n)={while(#A7522A007522_upto(N*3\2, N+1))); A7522[n]} \\ M. F. Hasler, May 22 2025

Equals A000040 INTERSECT A004215. - R. J. Mathar, Nov 22 2006

a(n) = 7 + A139487(n)*8, n >= 1. - Wolfdieter Lang, Feb 18 2015

A038872 Primes congruent to {0, 1, 4} mod 5.

Original entry on oeis.org

5, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, 211, 229, 239, 241, 251, 269, 271, 281, 311, 331, 349, 359, 379, 389, 401, 409, 419, 421, 431, 439, 449, 461, 479, 491, 499, 509, 521, 541, 569, 571, 599, 601, 619

Offset: 1

N. J. A. Sloane

Also odd primes p such that 5 is a square mod p: (5/p) = +1 for p > 5.
Primes of the form x^2 + x*y - y^2 (as well as of the form x^2 + 3*x*y + y^2), both with discriminant = 5 and class number = 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. [This was originally a separate entry, submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales, Jun 06 2008. R. J. Mathar proved on Jul 22 2008 that this coincides with the present sequence.]
Also primes of the form 5x^2 - y^2 (cf. A031363). - N. J. A. Sloane, May 30 2014
Also primes of the form x^2 + 4*x*y - y^2. Every binary quadratic primitive form of discriminant 20 or 5 has proper solutions for positive integers N given in A089270, including the present primes. Proof from computing the corresponding representative parallel primitive forms, which leads to x^2 - 5 == 0 (mod N) or x^2 + x - 1 == 0 (mod N) which have solutions precisely for these positive N values, including these primes. - Wolfdieter Lang, Jun 19 2019
For a Pythagorean triple a, b, c, these primes (and 2) are the possible prime factors of 2a + b, |2a - b|, 2b + a, and 2b - a. - J. Lowell, Nov 05 2011
The prime factors of A028387(n^2+3n+1). - Richard R. Forberg, Dec 12 2014
Except for p = 5, these are primes p that divide Fibonacci(p-1). - Dmitry Kamenetsky, Jul 27 2015
Apart from the first term, these are rational primes that decompose in the field Q[sqrt(5)]. For example, 11 = ((7 + sqrt(5))/2)*((7 - sqrt(5))/2), 19 = ((9 + sqrt(5))/2)*((9 - sqrt(5))/2). - Jianing Song, Nov 23 2018
The possible prime factors of x^2 - x - 1. - Charles R Greathouse IV, Mar 18 2022

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

T. D. Noe, Table of n, a(n) for n = 1..1000
C. Banderier, Calcul de (5/p)
Henri Darmon, Andrew Wiles's Marvelous Proof, Notices of the AMS (2017), Volume 64, Number 3 pp. 209-216. See p. 211.
Tamara M. Lavshuk, Regular polygons and polyhedra over finite field, Mathematical Notes of NEFU, Vol 22 No 4 (2015). Mentions this sequence.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A045468, A089270.
Cf. A038872 (d=5); A038873 (d=8); A068228, A141123 (d=12); A038883 (d=13). A038889 (d=17); A141111, A141112 (d=65).
Cf. A003631 (complement with respect to A000040).

Filtered(Concatenation([5],Flat(List([1..140],k->[5*k-1,5*k+1]))),IsPrime); # Muniru A Asiru, Nov 24 2018

[ p: p in PrimesUpTo(700) | p mod 5 in {0,1,4}]; // Vincenzo Librandi, Aug 21 2012

select(isprime, [5, seq(op([5*k-1,5*k+1]),k=1..1000)]); # Robert Israel, Dec 22 2014

Join[{5}, Select[Prime[Range[4, 100]], Mod[#, 5] == 1 || Mod[#, 5] == 4 &]] (* Alonso del Arte, Nov 27 2011 *)

forprime(p=2,1e3,if(kronecker(5,p)>=0,print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = A045468(n-1) for n > 1. - Robert Israel, Dec 22 2014

a(n) ~ 2n*log(n). - Charles R Greathouse IV, Nov 29 2016

Corrected and extended by Peter K. Pearson, May 29 2005

Edited by N. J. A. Sloane, Jul 28 2008 at the suggestion of R. J. Mathar

A038889 Primes p such that 17 is a square mod p.

Original entry on oeis.org

2, 13, 17, 19, 43, 47, 53, 59, 67, 83, 89, 101, 103, 127, 137, 149, 151, 157, 179, 191, 223, 229, 239, 251, 257, 263, 271, 281, 293, 307, 331, 349, 353, 359, 373, 383, 389, 409, 421, 433, 443, 457, 461, 463, 467, 491, 509, 523, 557, 563, 569, 577, 587, 593

Offset: 1

N. J. A. Sloane

nonn

Also primes of the form 2*x^2+x*y-2*y^2 (as well as of the form 2*x^2+5*x*y+y^2). Discriminant = 17. Class = 1. This was originally a separate entry, submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 06 2008. R. J. Mathar proved that this coincides with the present sequence, Jul 22 2008

Also, primes which are a square (mod 17) (or, (mod 34), cf. A191025). - M. F. Hasler, Jan 15 2016

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

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)
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A038889 (17 is a square mod p); A141111, A141112 (d=65).

Primes in A035258.

Select[Prime[Range[200]],JacobiSymbol[17,#]!=-1&] (* Harvey P. Dale, Sep 20 2011 *)

is(n)=isprime(n)&&issquare(Mod(17,n)) \\ Charles R Greathouse IV, Mar 21 2013

Edited by N. J. A. Sloane, Jul 28 2008 at the suggestion of R. J. Mathar

A038883 Odd primes p such that 13 is a square mod p.

Original entry on oeis.org

3, 13, 17, 23, 29, 43, 53, 61, 79, 101, 103, 107, 113, 127, 131, 139, 157, 173, 179, 181, 191, 199, 211, 233, 251, 257, 263, 269, 277, 283, 311, 313, 337, 347, 367, 373, 389, 419, 433, 439, 443, 467, 491, 503, 521, 523, 547, 563, 569, 571, 599, 601, 607, 641

Offset: 1

N. J. A. Sloane

nonn

Equivalently, by quadratic reciprocity (since 13 == 1 (mod 4)), primes p which are squares mod 13.
The squares mod 13 are 0, 1, 4, 9, 3, 12 and 10.
Also primes of the form x^2 + 3*x*y - y^2. Discriminant = 13. Class = 1. This was originally a separate entry, submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 06 2008. R. J. Mathar proved that this coincides with the present sequence, Jul 22 2008
Primes p such that x^2 + x = 3 has a solution mod p (the solutions over the reals are (-1+-sqrt(13))/2). - Joerg Arndt, Jul 27 2011

			13 == 1 (mod 3) and 1 is a square, so 3 is on the list.
101 is prime and congruent to 7^2 = 49 == 10 (mod 13), so 101 is on the list.

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Peter Luschny, Binary Quadratic Forms
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A038872 (d=5). A038873 (d=8). A068228, A141123 (d=12). A038883 (primes p such that d=13 is a square mod p). A038889 (d=17). A141111, A141112 (d=65).
Cf. A296937.
CF. A000040, A120330.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Select[ Prime@ Range@ 118, JacobiSymbol[ #, 13] > -1 &] (* Robert G. Wilson v, May 16 2008 *)
Select[Flatten[Table[13n + {1, 3, 4, 9, 10, 12}, {n, 50}]], PrimeQ[#] &] (* Alonso del Arte, Sep 16 2012 *)

forprime(p=3,1e3,if(issquare(Mod(13,p)),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

select( {is_A038883(n)=bittest(5659,n%13)&&isprime(n)}, [0..666]) \\ M. F. Hasler, Feb 17 2022

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([1, 3, -1])
print(Q.represented_positives(641, 'prime')) # Peter Luschny, Sep 20 2018

A000040 \ A120330 U {13}: Complement of A120330 in the primes, and 13. - M. F. Hasler, Feb 17 2022

Edited by N. J. A. Sloane, Apr 27 2008, Jul 28 2008

A141112 Primes of the form 2x^2+5xy-5y^2 (as well as of the form 7x^2+11xy+2y^2).

Original entry on oeis.org

2, 5, 7, 13, 37, 47, 67, 73, 83, 97, 137, 163, 167, 193, 197, 223, 227, 293, 307, 317, 353, 383, 397, 457, 463, 487, 557, 577, 587, 593, 613, 617, 643, 683, 733, 743, 773, 787, 827, 853, 863, 877, 947, 967, 977, 983, 1033, 1087, 1097, 1103, 1123, 1163, 1217, 1237, 1307, 1367, 1373

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 04 2008, Jun 05 2008

nonn

Both have discriminant = 65. Class = 2. 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(4)=37 because we can write 37=2*6^2+5*6*7-5*7^2 (or 37=7*1^2+11*1*2+2*2^2)

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

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)

Cf. A141111.

Subsequence of A243171.

Select[Prime[Range[250]], # == 5 || # == 13 || MatchQ[Mod[#, 65], Alternatives[2, 7, 8, 18, 28, 32, 33, 37, 47, 57, 58, 63]]&] (* Jean-François Alcover, Oct 28 2016 *)

# uses[binaryQF]
# The function binaryQF is defined in the link 'Binary Quadratic Forms'.
Q = binaryQF([2, 5, -5])
print(Q.represented_positives(1373, 'prime')) # Peter Luschny, Aug 12 2021

A033212 Primes congruent to 1 or 19 (mod 30).

Original entry on oeis.org

19, 31, 61, 79, 109, 139, 151, 181, 199, 211, 229, 241, 271, 331, 349, 379, 409, 421, 439, 499, 541, 571, 601, 619, 631, 661, 691, 709, 739, 751, 769, 811, 829, 859, 919, 991, 1009, 1021, 1039, 1051, 1069, 1129, 1171, 1201, 1231, 1249, 1279, 1291, 1321, 1381

Offset: 1

N. J. A. Sloane

Theorem: Same as primes of the form x^2+15*y^2 (discriminant -60). Proof: Cox, Cor. 2.27, p. 36.
Equivalently, primes congruent to 1 or 4 (mod 15). Also x^2+xy+4y^2 is the principal form of (fundamental) discriminant -15. The only other class for -15 contains the form 2x^2+xy+2y^2 (A106859), in the other genus. - Rick L. Shepherd, Jul 25 2014
Three further theorems (these were originally stated as conjectures, but are now known to be theorems, thanks to the work of J. B. Tunnell - see link):
1. The same as primes of the form x^2-xy+4y^2 (discriminant -15) and x^2-xy+19y^2 (discriminant -75), both with x and y nonnegative. - T. D. Noe, Apr 29 2008
2. The same as primes of the form x^2+xy+19y^2 (discriminant -75), with x and y nonnegative. - T. D. Noe, Apr 29 2008
3. The same as primes of the form x^2+5xy-5y^2 (discriminant 45). - N. J. A. Sloane, Jun 01 2014
Also primes of the form x^2+7*x*y+y^2 (discriminant 45).
Lemma (Will Jagy, Jun 12 2014): If c is any (positive or negative) even number, then x^2 + x y + c y^2 and x^2 + (4 c - 1) y^2 represent the same odd numbers.
Proof: x (x + y) + c y^2 = odd, therefore x is odd,  x + y odd, so y is even. Let y = 2 t. Then x( x + 2 t) + 4 c t^2 = x^2 + 2 x t  + 4 c t^2 = (x+t)^2 + (4c-1) t^2 = odd. QED With c = 4, neither one represents 2, so x^2+15y^2 and x^2+xy+4y^2 represent the same primes.
Also, primes which are squares (mod 3*5). Subsequence of A191018. - David Broadhurst and M. F. Hasler, Jan 15 2016

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
J. B. Tunnell, Proofs of Conjectures Concerning Entry A033212, the Sequence of Primes Congruent to 1 or 19 (mod 30)
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A106856, A139643, A106859.
Primes in A243173 and in A243174.
Cf. A141785 (d=45), A033212 (Primes of form x^2+15*y^2), A038872(d=5), A038873 (d=8), A068228, A141123 (d=12), A038883 (d=13), A038889 (d=17), A141111, A141112 (d=65).
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

QuadPrimes2[1, 0, 15, 10000] (* see A106856 *)
Select[Prime@Range[250], MemberQ[{1, 19}, Mod[#, 30]] &] (* Vincenzo Librandi, Apr 05 2015 *)

select(n->n%30==1||n%30==19, primes(100)) \\ Charles R Greathouse IV, Nov 09 2012

is(p)=issquare(Mod(p,15))&&isprime(p) \\ M. F. Hasler, Jan 15 2016

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

Edited by N. J. A. Sloane, Jun 01 2014 and Oct 18 2014: added Tunnell document, revised entry, merged with A141184. The latter entry was submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 12 2008.

Typo in crossrefs fixed by Colin Barker, Apr 05 2015

A141123 Primes of the form -x^2+2xy+2y^2 (as well as of the form 3x^2+6xy+2*y^2).

Original entry on oeis.org

2, 3, 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263, 311, 347, 359, 383, 419, 431, 443, 467, 479, 491, 503, 563, 587, 599, 647, 659, 683, 719, 743, 827, 839, 863, 887, 911, 947, 971, 983, 1019, 1031, 1091, 1103, 1151, 1163, 1187, 1223

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 05 2008

nonn

Discriminant = 12. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.

This is exactly {2} U A068231, primes congruent to 11 (mod 12). This is because the orders of imaginary quadratic fields with discriminant 12 has 1 class per genus (can be verified by the quadclassunit() function in PARI), so the primes represented by a binary quadratic form of this discriminant are determined by a congruence condition. - Jianing Song, Jun 22 2025

			a(3) = 11 because we can write 11 = -1^2 + 2*1*2 + 2*2^2 (or 11 = 3*1^2 + 6*1*1 + 2*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Robert Israel, Table of n, a(n) for n = 1..6514
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A038872 (d=5), A038873 (d=8), A068228 (d=12, 48, or -36), A038883 (d=13), A038889 (d=17), A141111 and A141112 (d=65).
Essentially the same as A068231 and A141187.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Cf. A084917.

N:= 2000:
S:= NULL:
for xx from 1 to floor(2*sqrt(N/3)) do
  for yy from ceil(sqrt(max(1,3*xx^2-N))) to floor(sqrt(3)*xx) do
     S:= S, 3*xx^2-yy^2;
od od:
sort(convert(select(isprime,{S}),list)); # Robert Israel, Jul 20 2020

Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -x^2 + 2*x*y + 2*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]]
(* or: *)
Select[Prime[Range[200]], # == 2 || # == 3 || Mod[#, 12] == 11&] (* Jean-François Alcover, Oct 25 2016, updated Oct 29 2016 *)

More terms from Colin Barker, Apr 05 2015

A141373 Primes of the form 3x^2+16y^2. Also primes of the form 4x^2+4xy-5y^2 (as well as primes the form 4x^2+12xy+3y^2).

Original entry on oeis.org

3, 19, 43, 67, 139, 163, 211, 283, 307, 331, 379, 499, 523, 547, 571, 619, 643, 691, 739, 787, 811, 859, 883, 907, 1051, 1123, 1171, 1291, 1459, 1483, 1531, 1579, 1627, 1699, 1723, 1747, 1867, 1987, 2011, 2083, 2131, 2179, 2203, 2251, 2347, 2371, 2467, 2539

Offset: 1

T. D. Noe, May 13 2005; Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 28 2008

nonn

The discriminant is -192 (or 96, or ...), depending on which quadratic form is used for the definition. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1. See A107132 for more information.

Except for 3, also primes of the forms 4x^2 + 4xy + 19y^2 and 16x^2 + 8xy + 19y^2. See A140633. - T. D. Noe, May 19 2008

			19 is a member because we can write 19=4*2^2+4*2*1-5*1^2 (or 19=4*1^2+12*1*1+3*1^2).

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

Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy] See Table II.
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. A107132, A139827, A107003, A141375, A141376 (d=96).
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),
A141336, A141337 (d=92),
A141338, A141339 (d=93),
A141161, A141163 (d=148),
A141165, A141166 (d=229),
A141167, A141167, A141167 (d=257).

[3] cat [ p: p in PrimesUpTo(3000) | p mod 24 in {19 } ]; // Vincenzo Librandi, Jul 24 2012

QuadPrimes2[3, 0, 16, 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)\16), if(isprime(t=w+16*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 09 2017

Except for 3, the primes are congruent to 19 (mod 24). - T. D. Noe, May 02 2008

More terms from Colin Barker, Apr 05 2015

Edited by N. J. A. Sloane, Jul 14 2019, combining two identical entries both with multiple cross-references.

cp's OEIS Frontend

A068228 Primes congruent to 1 (mod 12).

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A007519 Primes of form 8n+1, that is, primes congruent to 1 mod 8.

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A007522 Primes of the form 8n+7, that is, primes congruent to -1 mod 8.

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A038872 Primes congruent to {0, 1, 4} mod 5.

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A038889 Primes p such that 17 is a square mod p.

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A038883 Odd primes p such that 13 is a square mod p.

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A141112 Primes of the form 2x^2+5xy-5y^2 (as well as of the form 7x^2+11xy+2y^2).

A141123 Primes of the form -x^2+2xy+2y^2 (as well as of the form 3x^2+6xy+2*y^2).

A141373 Primes of the form 3x^2+16y^2. Also primes of the form 4x^2+4xy-5y^2 (as well as primes the form 4x^2+12xy+3y^2).