1,1

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

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).

Wolfdieter Lang, On Positive Integer Descartes-Steiner Curvature Quintuplets, arXiv:2503.08631 [math.NT], 2025. See pp. 23, 26-28.

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.

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

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

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

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

(PARI) 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

(PARI)

A007519_upto(N, start=1)=select(t->t%8==1, primes([start, N]))

#A7519=A007519_upto(10^5)

A007519(n)={while(#A7519<n, my(N=A7519[#A7519]); A7519=concat(A7519, A007519_upto(N*3\2, N+1))); A7519[n]} \\ M. F. Hasler, May 22 2025

(Haskell)

a007519 n = a007519_list !! (n-1)

a007519_list = filter ((== 1) . a010051) [1, 9..]

-- Reinhard Zumkeller, Mar 06 2012

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

(SageMath) # 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

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.

Sequence in context: A172280 A004625 A141174 * A163185 A138005 A267421

Adjacent sequences: A007516 A007517 A007518 * A007520 A007521 A007522

nonn,easy

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

approved