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

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

Author

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

Keywords

• nonn
• easy

Comments

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

References

• 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.

Links

• 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)

Crossrefs

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).

Programs

Haskell

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

Magma

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

Maple

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

Mathematica

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

PARI

(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.

PARI

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

Formula

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

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