cp's OEIS Frontend

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

%I A155560 #9 Jun 02 2025 01:18:59
%S A155560 13,37,52,61,73,97,100,109,117,148,157,169,181,193,208,229,241,244,
%T A155560 277,292,313,325,333,337,349,373,388,397,400,409,421,433,436,457,468,
%U A155560 481,541,549,577,592,601,613,628,637,657,661,673,676,709,724,733,757,769
%N A155560 Intersection of A000404 and A092572: N = a^2 + b^2 = c^2 + 3d^2 with a,b,c,d>0.
%C A155560 Nonsquare terms of A155563. - _Joerg Arndt_, Jan 11 2015
%e A155560 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).
%e A155560 a(2)=37 is the second smallest number which figures in A000404 and in A092572 as well.
%o A155560 (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}
%o A155560 for( n=1,10^3, isA155560(n) & print1(n","))
%o A155560 (PARI) is(n)=!issquare(n) && #bnfisintnorm(bnfinit(z^2+z+1), n) && #bnfisintnorm(bnfinit(z^2+1), n);
%o A155560 select(n->is(n), vector(1500,j,j)) \\ _Joerg Arndt_, Jan 11 2015
%K A155560 nonn
%O A155560 1,1
%A A155560 _M. F. Hasler_, Jan 24 2009