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A138320
Numerators of the difference between the squarefree totient analogs of the harmonic numbers and the harmonic numbers: F_n-H_n.
4
0, 1, 2, 5, 7, 4, 173, 587, 1481, 1859, 20701, 18391, 241393, 275713, 148367, 548423, 2342059, 241321, 41436061, 19263077, 40604659, 43779103, 1009564739, 1907583043, 9002492327, 9603126977, 27322095131, 25887926681, 752184042199
OFFSET
1,3
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' 3rd constant and gamma is Euler's constant.
FORMULA
a(n) = Numerator[(Sum_{k=1..n} mu^2(k)/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
Numerators 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[Numerator[Sum[MoebiusMu[k]^2/EulerPhi[k], {k, 1, n}]-HarmonicNumber[n]], {n, 1, 60}]
PROG
(PARI) for(n=1, 60, print1(numerator(sum(k=1, n, moebius(k)^2/eulerphi(k)) - sum(j=1, n, 1/j)), ", ")) \\ G. C. Greubel, Aug 31 2018
KEYWORD
frac,nonn
AUTHOR
Dick Boland (abstract(AT)imathination.org), Mar 14 2008, Mar 27 2008
STATUS
approved