cp's OEIS Frontend

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

%I A007522 M4376 #89 May 28 2025 01:06:33
%S A007522 7,23,31,47,71,79,103,127,151,167,191,199,223,239,263,271,311,359,367,
%T A007522 383,431,439,463,479,487,503,599,607,631,647,719,727,743,751,823,839,
%U A007522 863,887,911,919,967,983,991,1031,1039,1063,1087,1103,1151
%N A007522 Primes of the form 8n+7, that is, primes congruent to -1 mod 8.
%C A007522 Primes that are the sum of no fewer than four positive squares.
%C A007522 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 A007522 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 A007522 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 A007522 Subsequence of A141164. - _Reinhard Zumkeller_, Mar 26 2011
%C A007522 Also a subsequence of primes of the form x^2 + y^2 + z^2 + 1. - _Arkadiusz Wesolowski_, Apr 05 2012
%C A007522 Primes p such that p XOR 6 = p - 6. - _Brad Clardy_, Jul 22 2012
%D A007522 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
%D A007522 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007522 D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
%H A007522 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 A007522 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 A007522 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 A007522 Equals A000040 INTERSECT A004215. - _R. J. Mathar_, Nov 22 2006
%F A007522 a(n) = 7 + A139487(n)*8, n >= 1. - _Wolfdieter Lang_, Feb 18 2015
%p A007522 select(isprime, [seq(i,i=7..10000,8)]); # _Robert Israel_, Nov 22 2016
%t A007522 Select[8Range[200] - 1, PrimeQ] (* _Alonso del Arte_, Nov 07 2016 *)
%o A007522 (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.
%o A007522 (PARI)
%o A007522 A007522_upto(N, start=1)=select(p->p%8==7, primes([start, N]))
%o A007522 #A7522=A007522_upto(10^5)
%o A007522 A007522(n)={while(#A7522<n, my(N=A7522[#A7522]); A7522=concat(A7522, A007522_upto(N*3\2, N+1))); A7522[n]} \\ _M. F. Hasler_, May 22 2025
%o A007522 (Haskell)
%o A007522 a007522 n = a007522_list !! (n-1)
%o A007522 a007522_list = filter ((== 1) . a010051) a004771_list
%o A007522 -- _Reinhard Zumkeller_, Jan 29 2013
%o A007522 (Magma) [p: p in PrimesUpTo(2000) | p mod 8 eq 7]; // _Vincenzo Librandi_, Jun 26 2014
%Y A007522 Subsequence of A004771.
%Y A007522 Cf. A040098, A014754, A065907, A065908, A010051.
%Y A007522 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 A007522 nonn,easy
%O A007522 1,1
%A A007522 _N. J. A. Sloane_ and _Robert G. Wilson v_