cp's OEIS Frontend

A244019 Primes of the form 9x^2 + 6xy + 1849y^2.

%I A244019 #30 Oct 29 2020 15:18:47
%S A244019 1873,2017,2137,2377,2473,2689,3217,3529,3697,4057,4657,5569,6073,
%T A244019 6337,7177,7393,7417,7561,7681,7753,8017,8089,8233,8353,8737,8761,
%U A244019 9241,9601,9769,11113,11257,11617,12049,12433,12457,12721,13297,13633,13729,14281,15073,15313,16417,17977,19009,19273,20161,21169,23017,24049,25873,26161,26497,26713,29569,30097
%N A244019 Primes of the form 9x^2 + 6xy + 1849y^2.
%C A244019 Discriminant=-66528.
%C A244019 More than the usual number of terms are shown in order to display the difference from A139668 (Primes of the form x^2+1848y^2). The two sequences agree for the first 43 primes but then disagree [Jagy and Kaplansky].
%C A244019 This is a proper subsequence of A139668, since the terms of A244019 have the form z^2 + 1848*y^2: in fact, 9*x^2 + 6*x*y + 1849*y^2 = (3*x+y)^2 + 1848*y^2. [_Bruno Berselli_, Jun 20 2014]
%H A244019 Alois P. Heinz, <a href="/A244019/b244019.txt">Table of n, a(n) for n = 1..1000</a>
%H A244019 William C. Jagy and Irving Kaplansky, <a href="/A244019/a244019.pdf">Positive definite binary quadratic forms that represent the same primes</a> [Cached copy]
%H A244019 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 A244019 fd:=proc(a,b,c,M) local dd,xlim,ylim,x,y,t1,t2,t3,t4,i;
%p A244019 dd:=4*a*c-b^2;
%p A244019 if dd<=0 then error "Form should be positive definite."; break; fi;
%p A244019 t1:={};
%p A244019 xlim:=ceil( sqrt(M/a)*(1+abs(b)/sqrt(dd)));
%p A244019 ylim:=ceil( 2*sqrt(a*M/dd));
%p A244019 for x from 0 to xlim do
%p A244019 for y from -ylim to ylim do
%p A244019 t2 := a*x^2+b*x*y+c*y^2;
%p A244019 if t2 <= M then t1:={op(t1),t2}; fi; od: od:
%p A244019 t3:=sort(convert(t1,list));
%p A244019 t4:=[];
%p A244019 for i from 1 to nops(t3) do
%p A244019 if isprime(t3[i]) then t4:=[op(t4),t3[i]]; fi; od:
%p A244019 [[seq(t3[i],i=1..nops(t3))], [seq(t4[i],i=1..nops(t4))]];
%p A244019 end;
%p A244019 fd(9,6,1849,50000);
%t A244019 Reap[For[p = 2, p < 40000, p = NextPrime[p], s = Solve[x > 0 && 9 x^2 + 6 x y + 1849 y^2 == p, {x, y}, Integers]; If[s != {}, Print[p, " ", {x, y} /. s]; Sow[p]]]][[2, 1]] (* _Jean-François Alcover_, Oct 29 2020 *)
%Y A244019 Different from A139668 (Primes of the form x^2+1848y^2).
%K A244019 nonn
%O A244019 1,1
%A A244019 _N. J. A. Sloane_, Jun 19 2014