A155560 Intersection of A000404 and A092572: N = a^2 + b^2 = c^2 + 3d^2 with a,b,c,d>0.

13, 37, 52, 61, 73, 97, 100, 109, 117, 148, 157, 169, 181, 193, 208, 229, 241, 244, 277, 292, 313, 325, 333, 337, 349, 373, 388, 397, 400, 409, 421, 433, 436, 457, 468, 481, 541, 549, 577, 592, 601, 613, 628, 637, 657, 661, 673, 676, 709, 724, 733, 757, 769

Offset: 1

Author

M. F. Hasler, Jan 24 2009

Keywords

• nonn

Comments

Nonsquare terms of A155563. - Joerg Arndt, Jan 11 2015

Examples

			a(1)=13 is the least number that can be written as A+B and C+3D where A,B,C,D are positive squares (namely 13 = 2^2 + 3^2 = 1^2 + 3*2^2).
a(2)=37 is the second smallest number which figures in A000404 and in A092572 as well.
		

Programs

PARI

isA155560(n /* omit optional 2nd arg for the present sequence */, c=[3,1]) = { for(i=1,#c,for(b=1,sqrtint((n-1)\c[i]),issquare(n-c[i]*b^2)&next(2));return);1}
for( n=1,10^3, isA155560(n) & print1(n","))

PARI

is(n)=!issquare(n) && #bnfisintnorm(bnfinit(z^2+z+1), n) && #bnfisintnorm(bnfinit(z^2+1), n);
select(n->is(n), vector(1500,j,j)) \\ Joerg Arndt, Jan 11 2015