cp's OEIS Frontend

A385008 Integers k such that there exists an integer 0 such that sigma(m)^2 + sigma(k)^2 = 2*(m+k)^2.

%I A385008 #31 Jul 02 2025 00:58:16
%S A385008 4,284,1210,2924,4892,5564,6368,9962,10425,10856,13130,14595,18416,
%T A385008 28130,29631,35584,53296,53912,64617,66992,67268,71145,76084,86812,
%U A385008 87633,88730,100695,102364,104805,122390,123152,124155,139815,147610,153176,165596,168730,176336,180848
%N A385008 Integers k such that there exists an integer 0<m<k such that sigma(m)^2 + sigma(k)^2 = 2*(m+k)^2.
%C A385008 The numbers m and k form a PM(2,2)-amicable pair (PM = Power Mean). See Dimitrov link.
%C A385008 An amicable pair forms a PM(2,2)-amicable pair, so the larger member of an amicable pair A002046 is a term of this sequence.
%C A385008 From _David A. Corneth_, Jun 29 2025: (Start)
%C A385008 Terms <= 2500000 are 2, 3 or 4 (mod 6). Are there any terms from a different residue class?
%C A385008 m > ceiling(sqrt(2*k^2 + sigma(k)^2) - 2*k).
%C A385008 Proof: m + sigma(k)^2 < sigma(m)^2 + sigma(k)^2 = 2*(m+k)^2.
%C A385008 Solving m + sigma(k)^2 < 2*(m+k)^2 gives the desired result.
%C A385008 Also 8*k^2 > sigma(k)^2.
%C A385008 Proof: sigma(k)^2 < sigma(m)^2 + sigma(k)^2 = 2*(m+k)^2 < 2*(k+k)^2 = 8*k^2.
%C A385008 Combining the two we have 2*k^2 < sigma(k)^2 < 8*k^2.
%C A385008 In a search it helps to choose an odd prime p and then classify numbers m to points (m mod p, sigma(m) mod p).
%C A385008 Then if sigma(m)^2 + sigma(k)^2 = 2*(m+k)^2 then sigma(m)^2 == (2*(m+k)^2 - sigma(k)^2) (mod p).
%C A385008 for 0 <= m <= p-1 assume the equivalence class of m (mod p) which would give an equivalence class of sigma(m) mod p and reduce the numbers to be checked.  (End)
%H A385008 David A. Corneth, <a href="/A385008/b385008.txt">Table of n, a(n) for n = 1..154</a> (terms <= 2500000)
%H A385008 David A. Corneth, <a href="/A385008/a385008.gp.txt">PARI program</a>
%H A385008 S. I. Dimitrov, <a href="https://arxiv.org/abs/2408.07387">Generalizations of amicable numbers</a>, arXiv:2408.07387 [math.NT], 2024.
%e A385008 (1, 4) is such a pair because sigma(1)^2+sigma(4)^2 = 1^2+7^2 = 2*(1+4)^2.
%o A385008 (PARI) isok(k) = for (m=1, k-1, if (sigma(m)^2 + sigma(k)^2 == 2*(m+k)^2, return(m))); \\ _Michel Marcus_, Jun 15 2025
%Y A385008 Cf. A000203, A002046, A063990, A383484.
%K A385008 nonn
%O A385008 1,1
%A A385008 _S. I. Dimitrov_, Jun 15 2025
%E A385008 a(17)-a(26) from _Michel Marcus_, Jun 15 2025
%E A385008 a(27)-a(39) from _Michael S. Branicky_, Jun 26 2025