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 A385186 #9 Jun 26 2025 19:28:49 %S A385186 284,1210,2924,5564,6368,10856,14595,18416,76084,66992,71145,87633, %T A385186 88730,124155,139815,123152,153176,168730,176336,180848,203432,202444, %U A385186 365084,389924,430402,399592,455344,486178,514736,525915,669688,686072 %N A385186 Numbers y such that there exists an integer 0 < x < y such that sigma(x)^x * sigma(y)^y = (x+y)^(x+y). %C A385186 The numbers x and y form a WGM-amicable pair (WGM = Weighted Geometric Mean). See Dimitrov link. An amicable pair forms a WGM-amicable pair, so the larger member of an amicable pair A002046 is a term of this sequence. A check for x < y < 2*10^6 shows that all solutions (x,y) to the Diophantine equation sigma(x)^x * sigma(y)^y = (x+y)^(x+y) are amicable pairs. Although we cannot set an example, we cannot rule out the existence of WGM-amicable pairs that are not amicable pairs. %C A385186 It is important to note that the brute-force search should be performed on the logarithmic form of the equation: x*log(sigma(x)) + y*log(sigma(y)) = (x+y) * log(x+y). %H A385186 S. I. Dimitrov, <a href="https://arxiv.org/abs/2408.07387">Generalizations of amicable numbers</a>, arXiv:2408.07387 [math.NT], 2024. %e A385186 (220, 284) is such a pair because sigma(220)^220*sigma(284)^284 = (220+284)^(220+284). %Y A385186 Cf. A000203, A002046, A383932. %K A385186 nonn,hard %O A385186 1,1 %A A385186 _S. I. Dimitrov_, Jun 20 2025