A007520 Primes == 3 (mod 8).

3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187

Offset: 1

Author

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

Keywords

• nonn
• easy

Comments

Primes of the form 3x^2 + 2xy + 3y^2 with x and y in Z. - T. D. Noe, May 07 2005

Also, primes of the form X^2 + 2Y^2, X=|x-y|, Y=x+y. - Zak Seidov, Dec 06 2011

Each term is the sum of no fewer than three positive squares. - T. D. Noe, Nov 15 2010

Smallest terms expressible as sum of three distinct positive squares: 59 = 1^2 + 3^2 + 7^2, 83 = 3^2 + 5^2 + 7^2, 107, 131, 139, 179, 211, 227, 251, 283, 307. - Zak Seidov, Dec 06 2011

Except for the first term it appears that the terms of the sequence are also primes of the form 2k+1 such that 3*(2k+1) divides 2^k+1. - Hilko Koning, Dec 06 2019

From Hilko Koning, Nov 24 2021: (Start)

Theorem (Legendre symbol): With p an odd prime and a an integer coprime to p the Legendre symbol L(a/p) = -1 if a is a quadratic non-residue (mod p) and L(2/p) = -1 if p == +-3 (mod 8).

Theorem (Euler's criterion): L(a/p) == a^((p-1)/2) (mod p) so with a = 2 and prime p = 2k + 1 then -1 == 2^k (mod (2k+1)). So prime numbers 2k+1 = +-3 (mod 8) are the prime numbers 2k+1 | 2^k+1.

If 2k+1 == -3 (mod 8) then k is even and 2^k+1 is not divisible by 3 and if 2k+1 == +3 (mod 8) then k is odd and 2^k+1 is divisible by 3.

Hence prime numbers 2k+1 == 3 (mod 8) are prime numbers such that 3*(2k+1) | 2^k+1. Or, including the first term of the sequence, prime numbers 2k+1 with k odd such that 2k+1 | 2^k+1.

(End)

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Alexander Kalmynin, On Novák numbers, arXiv:1611.00417 [math.NT], 2016. See P0 in Theorem 7 p. 11.

Crossrefs

Cf. A294912, A045339 (for n >= 2).

Programs

Magma

[p: p in PrimesUpTo(2000) | p mod 8 eq 3]; // Vincenzo Librandi, Aug 07 2012

Maple

A007520 := proc(n)
    option remember;
    local a;
    if n = 1 then
        return 3;
    end if;
    a := nextprime(procname(n-1)) ;
    while modp(a,8) <> 3 do
        a := nextprime(a) ;
    end do:
    a ;
end proc:
seq(A007520(n),n=1..30) ; # R. J. Mathar, Apr 07 2017

Mathematica

lst={};Do[p=8*n+3;If[PrimeQ[p], AppendTo[lst, p]], {n, 0, 10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 22 2008 *)
p=3;k=0;nn=1000;Reap[While[kZak Seidov, Dec 06 2011 *)
Select[Prime[Range[200]],Mod[#,8]==3&] (* Harvey P. Dale, Apr 05 2023 *)

PARI

forprime(p=2,97,if(p%8==3,print1(p", "))) \\ Charles R Greathouse IV, Aug 17 2011