OFFSET
1,1
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.
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[k<nn, If[PrimeQ[p], k++; Sow[p]]; p=p+8]][[2, 1]] (* Zak Seidov, Dec 06 2011 *)
Select[Prime[Range[200]], Mod[#, 8]==3&] (* Harvey P. Dale, Apr 05 2023 *)
PROG
(PARI) forprime(p=2, 97, if(p%8==3, print1(p", "))) \\ Charles R Greathouse IV, Aug 17 2011
(Magma) [p: p in PrimesUpTo(2000) | p mod 8 eq 3]; // Vincenzo Librandi, Aug 07 2012
KEYWORD
nonn,easy
AUTHOR
STATUS
approved