A139645 - OEIS

%I #28 Sep 08 2022 08:45:33

%S 113,137,193,233,281,337,401,449,457,569,617,641,673,809,953,977,1009,

%T 1033,1129,1201,1289,1297,1409,1481,1801,1873,1913,2017,2081,2129,

%U 2137,2153,2297,2377,2417,2473,2521,2633,2657,2689,2713,2753,2801

%N Primes of the form x^2 + 112*y^2.

%C Discriminant is -448. See A139643 for more information.

%C Primes of the form 8*n + 1 which cannot be expressed as 7*k - 1, 7*k - 2, or 7*k - 4. a(n)^3 == 1 (mod 56). - _Gary Detlefs_, Jan 26 2014

%C The primes are congruent to {1, 9, 25} (mod 56).

%H Vincenzo Librandi and Ray Chandler, <a href="/A139645/b139645.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Vincenzo Librandi).

%H 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)

%p f:=n-> ceil((8*n+1)/7)-(8*n+1): for n from 1 to 350 do if isprime(8*n+1) and f(n)<>1 and f(n)<>2 and f(n)<>4 then print(8*n+1) fi od. # _Gary Detlefs_, Jan 26 2014

%t QuadPrimes2[1, 0, 112, 10000] (* see A106856 *)

%o (Magma) [ p: p in PrimesUpTo(3000) | p mod 56 in {1, 9, 25}]; // _Vincenzo Librandi_, Jul 28 2012

%o (Magma) k:=112; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // _Bruno Berselli_, Jun 01 2016

%K nonn,easy

%O 1,1

%A _T. D. Noe_, Apr 29 2008