This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383484 #42 Jun 02 2025 05:01:59
%S A383484 3,33,69,77,133,161,235,267,287,321,385,699,715,1235,1379,1437,1529,
%T A383484 1595,1653,1719,2047,2233,2241,2569,2727,2829,3237,3269,3999,4585,
%U A383484 4683,4911,5075,5163,5215,5497,5667,5691,7085,7089,7587,7761,7797,7945,8259,9159,9659,10653
%N A383484 Integers k such that there exists an integer 0<m<k such that sigma(m)^2 + sigma(k)^2 = (m+k)^2.
%C A383484 From _David A. Corneth_, May 04 2025: (Start)
%C A383484 If (t, u) is a divisor pair of sigma(k)^2 then m = (t + u - 2*k)/2, sigma(m) = m + k - t.
%C A383484 Proof:
%C A383484 Since sigma(m)^2 + sigma(k)^2 = (m+k)^2 we have sigma(k)^2 = (m+k)^2 - sigma(m)^2 = (m + k - sigma(m)) * (m + k + sigma(m)) = t * u where t, u | sigma(k)^2.
%C A383484 This gives the system (m + k - sigma(m)) = t and (m + k + sigma(m)) = u. Solving gives
%C A383484 m = (t + u - 2*k)/2, sigma(m) = m + k - t. For every pair (t, u) of divisors of sigma(k)^2 we can test if the given values of m and sigma(m) hold. If at least one of them holds then k is in the sequence. Q. E. D.
%C A383484 Are there any even terms? There are none in the first 1006 terms. (End)
%H A383484 David A. Corneth, <a href="/A383484/b383484.txt">Table of n, a(n) for n = 1..2003</a>
%H A383484 S. I. Dimitrov, <a href="https://arxiv.org/abs/2408.07387">Generalizations of amicable numbers</a>, arXiv:2408.07387 [math.NT], 2024.
%e A383484 (2, 3) is such a pair because sigma^2(2)+sigma^2(3) = 3^2+4^2 = (2+3)^2.
%e A383484 33 is in the sequence. As sigma(33)^2 = 2304 and for the divisor pair (32, 72) we have m = (32 + 72 - 2*33)/2 = 19 and sigma(m) = m + k - 32 = 19 + 33 - 32 = 20 and indeed sigma(19) = 20. - _David A. Corneth_, May 04 2025
%o A383484 (PARI) isok(k) = for (m=1, k, if (sigma(m)^2+sigma(k)^2==(m+k)^2, return(1))); \\ _Michel Marcus_, Apr 28 2025
%o A383484 (PARI) is(n) = {my(sn = sigma(n)^2, d = divisors(sn)); for(i = 1, #d / 2, k = (d[i] + d[#d + 1 - i] - 2*n) / 2; if(denominator(k) == 1, sk = n + k - d[i]; if(k < n && sigma(k) == sk, return(1)))); 0} \\ _David A. Corneth_, May 04 2025
%Y A383484 Cf. A063990, A259180, A383239.
%K A383484 nonn
%O A383484 1,1
%A A383484 _S. I. Dimitrov_, Apr 28 2025
%E A383484 More terms from _Michel Marcus_, Apr 28 2025