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A063947 Infinitary harmonic numbers: harmonic mean of infinitary divisors is an integer.

%I A063947 #24 Feb 16 2025 08:32:45
%S A063947 1,6,45,60,90,270,420,630,2970,5460,8190,9100,15925,27300,36720,40950,
%T A063947 46494,54600,81900,95550,136500,163800,172900,204750,232470,245700,
%U A063947 257040,409500,464940,491400,646425,716625,790398,791700,819000,900900
%N A063947 Infinitary harmonic numbers: harmonic mean of infinitary divisors is an integer.
%H A063947 Amiram Eldar, <a href="/A063947/b063947.txt">Table of n, a(n) for n = 1..239</a> (terms below 10^10)
%H A063947 P. Hagis, Jr. and G. L. Cohen, <a href="http://dx.doi.org/10.1017/S0004972700017949">Infinitary harmonic numbers</a>, Bull. Australian math. Soc., 41 (1990), 151-158 (Math. Rev. 91d:11001) (asymptotics).
%H A063947 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicMean.html">Harmonic Mean</a>
%H A063947 Wikipedia, <a href="http://www.wikipedia.org/wiki/Harmonic_mean">Harmonic mean</a>
%t A063947 bitty[ k_ ] := Union[ Flatten[ Outer[ Plus, Sequence @@ ({0, #} & /@ Union[ (2^Range[ 0, Floor[ Log[ 2, k ] ] ] ) Reverse[ IntegerDigits[ k, 2 ] ] ] ) ] ] ]; 1 + Flatten[ Position[ Table[ (Length[ # ] /(Plus @@ (1/#)) &)@ (Apply[ Times, (First[ it ] ^ (# /. z -> List)) ] & /@ Flatten[ Outer[ z, Sequence @@ (bitty /@ Last[ it = Transpose[ FactorInteger[ k ] ] ] ), 1 ] ]), {k, 2, 2^22 + 1} ], _Integer ] ] (* _Robert G. Wilson v_, Sep 04 2001 *)
%o A063947 (Haskell)
%o A063947 import Data.Ratio (denominator)
%o A063947 import Data.List (genericLength)
%o A063947 a063947 n = a063947_list !! (n-1)
%o A063947 a063947_list = filter ((== 1) . denominator . hm . a077609_row) [1..]
%o A063947    where hm xs = genericLength xs / sum (map (recip . fromIntegral) xs)
%o A063947 -- _Reinhard Zumkeller_, Jul 10 2013
%Y A063947 Cf. A037445, A049417.
%Y A063947 Cf. A077609.
%K A063947 nonn,nice
%O A063947 1,2
%A A063947 _Wouter Meeussen_, Sep 03 2001
%E A063947 More terms from _David W. Wilson_, Sep 04 2001