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A007522 Primes of the form 8n+7, that is, primes congruent to -1 mod 8.
(Formerly M4376)
78
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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)
FORMULA
Equals A000040 INTERSECT A004215. - R. J. Mathar, Nov 22 2006
a(n) = 7 + A139487(n)*8, n >= 1. - Wolfdieter Lang, Feb 18 2015
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 *)
PROG
(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)
(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
CROSSREFS
Subsequence of A004771.
Cf. A141174 (d = 32). A038872 (d = 5). A038873 (d = 8). A068228, A141123 (d = 12). A038883 (d = 13). A038889 (d = 17). A141111, A141112 (d = 65).
Sequence in context: A089199 A263874 A014663 * A141175 A295196 A287309
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 19 03:27 EDT 2024. Contains 370952 sequences. (Running on oeis4.)