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

%I #27 Jul 29 2023 06:35:52

%S 3,15,42,84,135,308,420,546,1428,1488,1890,2295,2660,3780,6210,7440,

%T 9424,12180,13392,18018,20832,24384,24570,43152,43400,64260,66960,

%U 77490,90090,98420,121920,127710,155610,200340,204600,227664,316992,348688,353400,461776,483210

%N Numbers whose divisors have a harmonic mean with a denominator 2.

%C Numbers k such that A099378(k) = 2.

%C The odd terms seem to be relatively rare: 3, 15, 135, 2295, 544635, 9258795, 22330035, 39118408875, ...

%C If k is in this sequence, then 2*k is in A348412.

%H Amiram Eldar, <a href="/A348411/b348411.txt">Table of n, a(n) for n = 1..310</a>

%e 3 is a term since the harmonic mean of its divisors, {1, 3}, is 3/2.

%e 15 is a term since the harmonic mean of its divisors, {1, 3, 5, 15}, is 5/2.

%p filter:= proc(n) local L,h;

%p L:= map(t->1/t,numtheory:-divisors(n));

%p denom(nops(L)/convert(L,`+`))=2;

%p end proc:

%p select(filter, [$1..10^6]); # _Robert Israel_, Oct 17 2021

%t Select[Range[10^5], Denominator[DivisorSigma[0, #]/DivisorSigma[-1, #]] == 2 &]

%t Select[Range[500000],Denominator[HarmonicMean[Divisors[#]]]==2&] (* _Harvey P. Dale_, Apr 06 2023 *)

%o (PARI) isok(m) = my(d=divisors(m)); denominator(#d/sum(k=1, #d, 1/d[k])) == 2; \\ _Michel Marcus_, Oct 18 2021

%o (Python)

%o from sympy import gcd, divisor_sigma

%o 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 Cf. A001599, A099377, A099378, A348412.

%Y Similar sequences: A159907, A330598.

%K nonn

%O 1,1

%A _Amiram Eldar_, Oct 17 2021