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%I M4376 #80 Sep 08 2022 08:44:35
%S 7,23,31,47,71,79,103,127,151,167,191,199,223,239,263,271,311,359,367,
%T 383,431,439,463,479,487,503,599,607,631,647,719,727,743,751,823,839,
%U 863,887,911,919,967,983,991,1031,1039,1063,1087,1103,1151
%N Primes of the form 8n+7, that is, primes congruent to -1 mod 8.
%C Primes that are the sum of no fewer than four positive squares.
%C 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.
%C 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
%C 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
%C Subsequence of A141164. - _Reinhard Zumkeller_, Mar 26 2011
%C Also a subsequence of primes of the form x^2 + y^2 + z^2 + 1. - _Arkadiusz Wesolowski_, Apr 05 2012
%C Primes p such that p XOR 6 = p - 6. - _Brad Clardy_, Jul 22 2012
%D Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
%H Ray Chandler, <a href="/A007522/b007522.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H Milton Abramowitz and Irene A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%F Equals A000040 INTERSECT A004215. - _R. J. Mathar_, Nov 22 2006
%F a(n) = 7 + A139487(n)*8, n >= 1. - _Wolfdieter Lang_, Feb 18 2015
%p select(isprime, [seq(i,i=7..10000,8)]); # _Robert Israel_, Nov 22 2016
%t Select[8Range[200] - 1, PrimeQ] (* _Alonso del Arte_, Nov 07 2016 *)
%o (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)
%o (Haskell)
%o a007522 n = a007522_list !! (n-1)
%o a007522_list = filter ((== 1) . a010051) a004771_list
%o -- _Reinhard Zumkeller_, Jan 29 2013
%o (Magma) [p: p in PrimesUpTo(2000) | p mod 8 eq 7]; // _Vincenzo Librandi_, Jun 26 2014
%Y Subsequence of A004771.
%Y Cf. A040098, A014754, A065907, A065908, A010051.
%Y Cf. A141174 (d = 32). A038872 (d = 5). A038873 (d = 8). A068228, A141123 (d = 12). A038883 (d = 13). A038889 (d = 17). A141111, A141112 (d = 65).
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, _Robert G. Wilson v_
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