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Positive integers k such that the numerator of the harmonic mean of the proper divisors of k is equal to k.
3

%I #19 Mar 02 2020 08:54:02

%S 4,27,28,54,56,64,68,91,99,100,133,138,148,154,165,188,217,222,247,

%T 259,268,276,279,290,301,308,369,375,388,403,414,427,428,430,469,474,

%U 481,508,511,540,544,548,549,553,559,589,609,621,627,628,639,642,665,668

%N Positive integers k such that the numerator of the harmonic mean of the proper divisors of k is equal to k.

%H Amiram Eldar, <a href="/A250095/b250095.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..2000 from Colin Barker)

%e 27 is a term because the proper divisors of 27 are [1,3,9] and 3 / (1/1 + 1/3 + 1/9) = 27/13.

%t Select[Range[668], CompositeQ[#] && Numerator[(DivisorSigma[0, #] - 1) * #/(DivisorSigma[1, #] - 1)] == # &] (* _Amiram Eldar_, Mar 02 2020 *)

%o (PARI)

%o harmonicmean(v) = #v / sum(k=1, #v, 1/v[k])

%o properdivisors(n) = d=divisors(n); vector(#d-1, k, d[k])

%o s=[]; for(n=2, 1000, if(numerator(harmonicmean(properdivisors(n)))==n, s=concat(s, n))); s

%Y Cf. A247081, A250094.

%K nonn

%O 1,1

%A _Colin Barker_, Nov 12 2014