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 A385492 #27 Jul 31 2025 08:41:09 %S A385492 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89, %T A385492 97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179, %U A385492 181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,284,293,307,311,313 %N A385492 Numbers y such that there exists an integer 0 < x < y such that x^sigma(x) * y^sigma(y) = (x*y)^(x+y). %C A385492 The numbers x and y form an IWGM-amicable pair (IWGM = Inverse Weighted Geometric Mean). See Dimitrov link. %C A385492 Every amicable pair forms an IWGM-amicable pair, so the larger member of an amicable pair A002046 is a term of this sequence. In addition, every pair of the form (1, p), where p is a prime, also forms an IWGM-amicable pair, so every term of A000040 is a term of this sequence. %C A385492 Most likely, there are no other solutions except those whose second component belongs to A000040 or A002046. %H A385492 Robert Israel, <a href="/A385492/b385492.txt">Table of n, a(n) for n = 1..1000</a> %H A385492 S. I. Dimitrov, <a href="https://arxiv.org/abs/2408.07387">Generalizations of amicable numbers</a>, arXiv:2408.07387 [math.NT], 2024. %e A385492 (220, 284) is such a pair because 220^sigma(220)*284^sigma(284) = (220*284)^(220+284). %p A385492 S:= map(numtheory:-sigma, [$1..1000]): %p A385492 select(y -> ormap(x -> x^(S[x]-x-y) * y^(S[y]-x-y) = 1, [$1..y-1]), [$1..1000]); # _Robert Israel_, Jul 30 2025 %o A385492 (PARI) isok(y) = my(z=y^sigma(y)); for (x=1, y-1, if (z*x^sigma(x) == (x*y)^(x+y), return(1));); \\ _Michel Marcus_, Jul 02 2025 %Y A385492 Cf. A000203, A000040, A002046, A385186. %K A385492 nonn %O A385492 1,1 %A A385492 _S. I. Dimitrov_, Jun 30 2025