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

1,1

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

Links

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

Example

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.

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 *)

Prog

(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, ", ")))

Crossrefs

Cf. A000404, A003136, A068228, A084916.

Sequence in context: A075033 A017533 A309809 * A298483 A344900 A195558

Adjacent sequences: A264729 A264730 A264731 * A264733 A264734 A264735

Keyword

nonn

Author

Altug Alkan, Nov 22 2015

Status

approved