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A175505 Numerator of A053818(n)/A023896(n) = antiharmonic mean of numbers k such that gcd(k,n) = 1, 1 <= k < n.

%I A175505 #30 May 23 2025 08:50:40
%S A175505 1,1,5,5,3,13,13,21,53,7,7,49,25,29,31,85,11,109,37,27,43,15,15,193,
%T A175505 83,53,485,113,19,59,61,341,67,23,71,433,73,77,79,107,27,83,85,59,271,
%U A175505 31,31,769,685,167,103,209,35,973,37,449,115,39,39,239,121,125,379,1365
%N A175505 Numerator of A053818(n)/A023896(n) = antiharmonic mean of numbers k such that gcd(k,n) = 1, 1 <= k < n.
%C A175505 See A175506 - denominators of the antiharmonic means B of numbers k such that gcd(k, n) = 1 for numbers n >= 1 and k < n where B = A053818(n) / A023896(n) = a(n) / A175506(n).
%H A175505 Wikipedia, <a href="http://en.wikipedia.org/wiki/Contraharmonic_mean">Contraharmonic mean</a>.
%F A175505 a(n) = A053818(n) * A175506(n) / A023896(n).
%F A175505 Sum_{k=1..n} a(k)/A175506(k) ~ n^2/3. - _Amiram Eldar_, Dec 07 2023
%p A175505 antiHMean := proc(L)
%p A175505     add(i^2,i=L)/add(i,i=L) ;
%p A175505 end proc:
%p A175505 A175505 := proc(n)
%p A175505     local kset,k ;
%p A175505     kset := [1] ;
%p A175505     for k from 2 to n do
%p A175505         if igcd(k,n) = 1 then
%p A175505             kset := [op(kset),k] ;
%p A175505         end if;
%p A175505     end do:
%p A175505     antiHMean(kset) ;
%p A175505     numer(%) ;
%p A175505 end proc: # _R. J. Mathar_, Sep 26 2013
%t A175505 f[n_] := 2Plus @@ (Select[ Range@n, GCD[ #, n] == 1 &]^2)/(n*EulerPhi@n); f[1] = 1; Numerator@Array[f, 65] (* _Robert G. Wilson v_, Jul 01 2010 *)
%o A175505 (PARI) A175505(n)=numerator((2*n+(-1)^omega(n)*A007947(n)/n)/3) \\ _M. F. Hasler_, Nov 29 2010
%o A175505 (PARI) a(n) = {my(f = factor(n)); numerator(if(n == 1, 1, 2*n/3 + (1/3) * prod(i = 1, #f~, 1 - f[i, 1])/eulerphi(f)));} \\ _Amiram Eldar_, Dec 07 2023
%Y A175505 Cf. A023896, A053818, A175506 (denominators).
%K A175505 nonn,frac
%O A175505 1,3
%A A175505 _Jaroslav Krizek_, May 31 2010, Jun 01 2010
%E A175505 More terms from _Robert G. Wilson v_, Jul 01 2010