A071011 Numbers n such that n is a sum of 2 squares (i.e., n is in A001481(k)) and sigma(n) == 0 (mod 4).

65, 85, 125, 130, 145, 170, 185, 205, 221, 250, 260, 265, 290, 305, 340, 365, 370, 377, 410, 442, 445, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 580, 585, 610, 629, 680, 685, 689, 697, 730, 740, 745, 754, 765, 785, 793, 820, 865, 884, 890, 901, 905

Offset: 1

Author

Benoit Cloitre, May 19 2002

Keywords

• easy
• nonn

Comments

It is conjectured that if m is not a sum of 2 squares (i.e., m is in A022544(k)) sigma(m) == 0 (mod 4).

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001481, A022544.

Programs

Mathematica

Select[Range[10^3], And[SquaresR[2, #] > 0, Divisible[DivisorSigma[1, #], 4]] &] (* Michael De Vlieger, Jul 30 2017 *)

PARI

for(n=1,1000,if(1-sign(sum(i=0,n,sum(j=0,i,if(i^2+j^2-n,0,1))))+sigma(n)%4==0,print1(n,",")))

Python

from math import prod
from itertools import count, islice
from sympy import factorint
def A071011_gen(): # generator of terms
    return filter(lambda n:(lambda f:all(p & 3 != 3 or e & 1 == 0 for p, e in f) and prod((p**(e+1)-1)//(p-1) & 3 for p, e in f) & 3 == 0)(factorint(n).items()),count(0))
A071011_list = list(islice(A071011_gen(),30)) # Chai Wah Wu, Jun 27 2022