A264732 Löschian numbers (A003136) which are the sum of 2 nonzero squares.

13, 25, 37, 52, 61, 73, 97, 100, 109, 117, 148, 157, 169, 181, 193, 208, 225, 229, 241, 244, 277, 289, 292, 313, 325, 333, 337, 349, 373, 388, 397, 400, 409, 421, 433, 436, 457, 468, 481, 541, 549, 577, 592, 601, 613, 625, 628, 637, 657, 661, 673, 676, 709, 724, 733

Offset: 1

Author

Altug Alkan, Nov 22 2015

Keywords

• nonn

Comments

n is in the sequence iff 4*n is.

If a(n) is a prime number, a(n) mod 12 = 1.

Prime terms of sequence are listed in A068228 that lists generalized cuban primes (A007645) which are the sum of 2 nonzero squares.

Also positive numbers of the form x^2 - 3*y^2 (A084916) that are the sum of 2 nonzero squares. - Frank M Jackson, Oct 13 2019

Examples

			a(1) = 13 because 13 = 3^2 + 3*1 + 1^2 = 3^2 + 2^2.
a(2) = 25 because 25 = 5^2 + 5*0 + 0^2 = 4^2 + 3^2.
a(3) = 37 because 37 = 4^2 + 4*3 + 3^2 = 6^2 + 1^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000404, A003136, A068228, A084916.

Programs

Mathematica

Select[Range@750, Length[PowersRepresentations[#, 2, 2] /. {0, }->Nothing]>0 && Reduce[#==x^2+x*y+y^2, {x, y}, Integers]=!=False &] (* _Frank M Jackson, Oct 13 2019 *)

PARI

isok(n) = { for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
is(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
for(n=1, 1e3, if( is(n) && isok(n), print1(n, ", ")))