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A007522 Primes of the form 8n+7, that is, primes congruent to -1 mod 8.
(Formerly M4376)
69
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 K├Ârper, Springer, 1981.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

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. 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).

Sequence in context: A089199 A263874 A014663 * A141175 A275777 A157811

Adjacent sequences:  A007519 A007520 A007521 * A007523 A007524 A007525

KEYWORD

nonn,easy,changed

AUTHOR

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

STATUS

approved

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Last modified December 3 01:12 EST 2016. Contains 278694 sequences.