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A348411 Numbers whose divisors have a harmonic mean with a denominator 2.

%I A348411 #27 Jul 29 2023 06:35:52
%S A348411 3,15,42,84,135,308,420,546,1428,1488,1890,2295,2660,3780,6210,7440,
%T A348411 9424,12180,13392,18018,20832,24384,24570,43152,43400,64260,66960,
%U A348411 77490,90090,98420,121920,127710,155610,200340,204600,227664,316992,348688,353400,461776,483210
%N A348411 Numbers whose divisors have a harmonic mean with a denominator 2.
%C A348411 Numbers k such that A099378(k) = 2.
%C A348411 The odd terms seem to be relatively rare: 3, 15, 135, 2295, 544635, 9258795, 22330035, 39118408875, ...
%C A348411 If k is in this sequence, then 2*k is in A348412.
%H A348411 Amiram Eldar, <a href="/A348411/b348411.txt">Table of n, a(n) for n = 1..310</a>
%e A348411 3 is a term since the harmonic mean of its divisors, {1, 3}, is 3/2.
%e A348411 15 is a term since the harmonic mean of its divisors, {1, 3, 5, 15}, is 5/2.
%p A348411 filter:= proc(n) local L,h;
%p A348411   L:= map(t->1/t,numtheory:-divisors(n));
%p A348411   denom(nops(L)/convert(L,`+`))=2;
%p A348411 end proc:
%p A348411 select(filter, [$1..10^6]); # _Robert Israel_, Oct 17 2021
%t A348411 Select[Range[10^5], Denominator[DivisorSigma[0, #]/DivisorSigma[-1, #]] == 2 &]
%t A348411 Select[Range[500000],Denominator[HarmonicMean[Divisors[#]]]==2&] (* _Harvey P. Dale_, Apr 06 2023 *)
%o A348411 (PARI) isok(m) = my(d=divisors(m)); denominator(#d/sum(k=1, #d, 1/d[k])) == 2; \\ _Michel Marcus_, Oct 18 2021
%o A348411 (Python)
%o A348411 from sympy import gcd, divisor_sigma
%o A348411 A348411_list = [n for n in range(1,10**3) if (lambda x, y: 2*gcd(x,y*n)==x)(divisor_sigma(n),divisor_sigma(n,0))] # _Chai Wah Wu_, Oct 20 2021
%Y A348411 Cf. A001599, A099377, A099378, A348412.
%Y A348411 Similar sequences: A159907, A330598.
%K A348411 nonn
%O A348411 1,1
%A A348411 _Amiram Eldar_, Oct 17 2021