OFFSET
1,2
COMMENTS
F_n - H_n approaches a constant, 'kappa', conjectured to be equivalent to the difference of B_3-gamma, where B_3 is Mertens's 3rd constant and gamma is Euler's constant.
Original data was given as {1, 1, 12, 24, 240, 80, 560, 3360, 30240, 7560, 831600, 831600, 93600, 21621600, 6177600, 12355200, 2940537600, 980179200, 55870214400, 2234808576, 3724680960, 177365760, 49597067520, 29758240512, 3719780064000} which is in error for this sequence. - G. C. Greubel, Sep 14 2018
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = denominator( (Sum_{k=1..n} mu(k)^2/phi(k)) - H_n) where mu(k) is the Mobius function, phi(k) is Euler's Totient function and H_n is the n-th Harmonic Number.
EXAMPLE
Denominators of F_n - H_n, e.g., -F_1 - H_1 = (1/1 - 1/1), F_2 = ((1/1 - 1/1) + (1/1 - 1/2)), ...
F_11 = ((1/1 - 1/1) + (1/1 - 1/2) + (1/2 - 1/3) + (0 - 1/4) + (1/4 - 1/5) + (1/2 - 1/6) + (1/6 - 1/7) + (0 - 1/8) + (0 - 1/9) + (1/4 - 1/10) + (1/10 - 1/11)).
MATHEMATICA
Table[Denominator[Sum[MoebiusMu[k]^2/EulerPhi[k], {k, 1, n}]-HarmonicNumber[n]], {n, 1, 60}]
PROG
(PARI) for(n=1, 60, print1(denominator(sum(k=1, n, moebius(k)^2/eulerphi(k) ) - sum(j=1, n, 1/j)), ", ")) \\ G. C. Greubel, Sep 14 2018
CROSSREFS
KEYWORD
frac,nonn
AUTHOR
Dick Boland (abstract(AT)imathination.org), Mar 14 2008, Mar 27 2008
EXTENSIONS
Data replaced by G. C. Greubel, Sep 14 2018
STATUS
approved