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A384706 Integers y such that there exists an integer 0 < x < y such that y/sigma(x) + x/sigma(y) = 1.

%I A384706 #23 Jul 10 2025 14:01:51
%S A384706 14,20,42,54,62,88,99,108,114,124,126,132,189,195,204,210,220,238,252,
%T A384706 254,272,284,328,340,385,414,420,432,455,464,468,495,508,528,560,572,
%U A384706 608,621,630,663,693,748,828,837,870,888,1008,1089,1136,1192,1197,1210,1288,1416,1422,1440
%N A384706 Integers y such that there exists an integer 0 < x < y such that y/sigma(x) + x/sigma(y) = 1.
%C A384706 Let sigma be the usual sum-of-divisors function. We say that x and y form a cross-harmonious pair if y/sigma(x) + x/sigma(y) = 1. An amicable pair forms a cross-harmonious pair, so the larger member of an amicable pair A002046 is a term of this sequence.
%C A384706 If a < b are Mersenne exponents (A000043) then 2^(a-1) * (2^b - 1) is a term, as it and 2^(b-1) * (2^a - 1) are a cross-harmonious pair. - _Robert Israel_, Jul 07 2025
%H A384706 Robert Israel, <a href="/A384706/b384706.txt">Table of n, a(n) for n = 1..500</a>
%H A384706 S. I. Dimitrov, <a href="https://arxiv.org/abs/2408.07387">Generalizations of amicable numbers</a>, arXiv:2408.07387 [math.NT], 2024.
%e A384706 12 and 14 form a cross-harmonious pair since 14/sigma(12) + 12/sigma(14) = 14/28 + 12/24 = 1.
%p A384706 N:= 10000: # for terms <= N
%p A384706 S:= map(numtheory:-sigma, [$1..N]):
%p A384706 filter:= proc(x) ormap(y -> y/S[x]+x/S[y]=1, [$1..x-1]) end proc:
%p A384706 select(filter, [$1..N]); # _Robert Israel_, Jul 07 2025
%o A384706 (PARI) isok(y) = for (x=1, y-1, if (y/sigma(x) + x/sigma(y) == 1, return(x));); \\ _Michel Marcus_, Jun 09 2025
%Y A384706 Cf. A000043, A000203, A000396, A002025, A002046, A253535, A253534.
%K A384706 nonn
%O A384706 1,1
%A A384706 _S. I. Dimitrov_, Jun 07 2025