login
Denominators of the squarefree totient analogs of the harmonic numbers F_n.
6

%I #11 Aug 28 2018 15:00:54

%S 1,1,2,2,4,4,12,12,12,3,30,30,20,60,120,120,240,240,720,720,720,720,

%T 7920,7920,7920,7920,7920,7920,55440,55440,55440,55440,55440,27720,

%U 3465,3465,4620,13860,27720,27720,13860,6930,3465,3465,3465,6930,79695,79695

%N Denominators of the squarefree totient analogs of the harmonic numbers F_n.

%C F_n-H_n approaches a constant, 'kappa', conjectured to be equivalent to the difference of B_3-gamma, where B_3 is Mertens' 3rd constant and gamma is Euler's constant.

%H G. C. Greubel, <a href="/A138317/b138317.txt">Table of n, a(n) for n = 1..10000</a>

%H Dick Boland, <a href="http://www.imathination.org/kappa/kappa.pdf">An Analog of the Harmonic Numbers Over the Squarefree Integers</a>

%F a(n)=Denominator[sum(k=1 to n)mu^2(k)/phi(k)] where mu(k) is the Mobius function and phi(k) is Euler's Totient function.

%e Denominators of F_n, e.g., - F_1 = (1/1), F_2 = (1/1 + 1/1), ... F_11 = (1/1 + 1/1 + 1/2 + 0 + 1/4 + 1/2 + 1/6 + 0 + 0 + 1/4 + 1/10).

%t Table[Denominator[Sum[MoebiusMu[k]^2/EulerPhi[k], {k, 1, n}]], {n, 1, 60}]

%o (PARI) a(n) = denominator(sum(k=1, n, if (issquarefree(k), 1/eulerphi(k)))); \\ _Michel Marcus_, Aug 28 2018

%Y Cf. A138312, A138313, A138312, A138316, A138320, A138321, A083343, A001620.

%K frac,nonn

%O 1,3

%A Dick Boland (abstract(AT)imathination.org), Mar 13 2008, Mar 27 2008