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

1, 1, 5, 5, 3, 13, 13, 21, 53, 7, 7, 49, 25, 29, 31, 85, 11, 109, 37, 27, 43, 15, 15, 193, 83, 53, 485, 113, 19, 59, 61, 341, 67, 23, 71, 433, 73, 77, 79, 107, 27, 83, 85, 59, 271, 31, 31, 769, 685, 167, 103, 209, 35, 973, 37, 449, 115, 39, 39, 239, 121, 125, 379, 1365

Offset: 1

Author

Jaroslav Krizek, May 31 2010, Jun 01 2010

Keywords

• nonn
• frac

Comments

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).

Links

• Wikipedia, Contraharmonic mean.

Crossrefs

Cf. A023896, A053818, A175506 (denominators).

Programs

Maple

antiHMean := proc(L)
    add(i^2,i=L)/add(i,i=L) ;
end proc:
A175505 := proc(n)
    local kset,k ;
    kset := [1] ;
    for k from 2 to n do
        if igcd(k,n) = 1 then
            kset := [op(kset),k] ;
        end if;
    end do:
    antiHMean(kset) ;
    numer(%) ;
end proc: # R. J. Mathar, Sep 26 2013

Mathematica

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 *)

PARI

A175505(n)=numerator((2*n+(-1)^omega(n)*A007947(n)/n)/3) \\ M. F. Hasler, Nov 29 2010

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

Formula

a(n) = A053818(n) * A175506(n) / A023896(n).

Sum_{k=1..n} a(k)/A175506(k) ~ n^2/3. - Amiram Eldar, Dec 07 2023

Extensions

More terms from Robert G. Wilson v, Jul 01 2010