cp's OEIS Frontend

A384814 Integers k such that there exists an integer 0 such that (1/sigma(m) + 1/sigma(k))*(m+k) = 2.

%I A384814 #13 Jul 25 2025 02:18:22
%S A384814 28,56,66,88,90,114,132,174,220,234,238,246,284,306,308,312,340,348,
%T A384814 356,496,532,534,552,618,620,654,668,728,752,760,786,812,856,963,990,
%U A384814 992,996,1052,1092,1148,1180,1196,1210,1220,1232,1244,1288,1320,1326,1364,1372,1474,1494,1580
%N A384814 Integers k such that there exists an integer 0<m<k such that (1/sigma(m) + 1/sigma(k))*(m+k) = 2.
%C A384814 The numbers m and k form a HM(1,2)-amicable pair (HM = harmonic mean). See Dimitrov link. An amicable pair forms a HM(1,2)-amicable pair, so the larger member of an amicable pair A002046 is a term of this sequence.
%H A384814 Robert Israel, <a href="/A384814/b384814.txt">Table of n, a(n) for n = 1..250</a>
%H A384814 S. I. Dimitrov, <a href="https://arxiv.org/abs/2408.07387">Generalizations of amicable numbers</a>, arXiv:2408.07387 [math.NT], 2024.
%e A384814 (20, 28) is such a pair because (1/sigma(20) + 1/sigma(28))*(20+28) = 2.
%p A384814 S:= map(numtheory:-sigma,[$1..3000]):
%p A384814 filter:= proc(k)
%p A384814    ormap(m -> (1/S[m] + 1/S[k])*(m+k) = 2, [$1..k-1])
%p A384814 end proc:
%p A384814 select(filter, [$1..3000]); # _Robert Israel_, Jul 25 2025
%o A384814 (PARI) isok(k) = for(m=1, k-1, if ((1/sigma(m) + 1/sigma(k))*(m+k) == 2, return(1))); \\ _Michel Marcus_, Jun 10 2025
%Y A384814 Cf. A000203, A000396, A002025, A002046, A383964.
%K A384814 nonn
%O A384814 1,1
%A A384814 _S. I. Dimitrov_, Jun 10 2025