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A385356 Numbers x such that there exist two integers 0 and z>0 such that sigma(x)^2 = sigma(y)^2 = x^2 + y^2 + z^2.

%I A385356 #25 Jul 26 2025 20:24:35
%S A385356 2,40,164,196,224,1120,3040,13440,22932,44200,76160,90848,91720,
%T A385356 174592,530200,619840,687184,872960,1686400,1767040,1807120,1927680,
%U A385356 1990912,2154880,3653760,4286880,5637632,5759680,6442128,8225280,8943800,9264320,9465600,9694080
%N A385356 Numbers x such that there exist two integers 0<x<=y and z>0 such that sigma(x)^2 = sigma(y)^2 = x^2 + y^2 + z^2.
%C A385356 The numbers x, y and z form a sigma-quadratic triple. See Dimitrov link.
%H A385356 Chai Wah Wu, <a href="/A385356/b385356.txt">Table of n, a(n) for n = 1..161</a>
%H A385356 S. I. Dimitrov, <a href="https://arxiv.org/abs/2408.07387">Generalizations of amicable numbers</a>, arXiv:2408.07387 [math.NT], 2024.
%e A385356 (40, 58, 56) is such a triple because sigma(40)^2 = sigma(58)^2 = 90^2 = 40^2 + 58^2 + 56^2.
%o A385356 (Python)
%o A385356 from itertools import count, islice
%o A385356 from sympy import divisor_sigma
%o A385356 from sympy.ntheory.primetest import is_square
%o A385356 def A385356_gen(startvalue=1): # generator of terms >= startvalue
%o A385356     for x in count(max(startvalue,1)):
%o A385356         sx, x2 = int(divisor_sigma(x)), x**2
%o A385356         sx2 = sx**2
%o A385356         if sx2>x2:
%o A385356             for y in count(x):
%o A385356                 if (k:=sx2-x2-y**2)<=0:
%o A385356                     break
%o A385356                 if is_square(k) and sx==divisor_sigma(y):
%o A385356                     yield x
%o A385356                     break
%o A385356 A385356_list = list(islice(A385356_gen(),8)) # _Chai Wah Wu_, Jul 02 2025
%Y A385356 Cf. A000203, A066784, A096907, A096908, A096909, A096910, A385325.
%K A385356 nonn
%O A385356 1,1
%A A385356 _S. I. Dimitrov_, Jun 26 2025
%E A385356 Data corrected by _David A. Corneth_, Jun 27 2025
%E A385356 a(18)-a(34) from _Chai Wah Wu_, Jul 02 2025