OFFSET
0,3
FORMULA
From Thomas Ordowski, Mar 21 2023: (Start)
Sum_{k=1..n} 1/H(k) ~ Sum_{k=2..n} 1/log(k) ~ Integral_{2..n} dx/log(x) = Li(n).
Sum_{k=1..n} 1/H(k) = Sum_{k=1..n} 1/(log(k + 1/2) + gamma) - C + o(1), where gamma = A001620 = 0.577... is Euler's constant and the constant C = 0.0229825...
Sum_{k=1..n} 1/H(k) = exp(-gamma)*(Ei(log(n) + gamma) - 1) + o(1), where Ei(x) is the exponential integral function of real x, and we have Ei(log(x)) = li(x).
Sum_{k=1..n} 1/H(k) = n/(H(n) - 1 + ...) = n/(log(n) + gamma - 1 + O(1/log(n))).
Theorem: lim_{n->oo} (H(n) - n / Sum_{k=1..n} 1/H(k)) = 1, see my third formula.
Proof: since Integral dx / (log(x) + gamma) = exp(-gamma)*Ei(log(x) + gamma) + c, so we get lim_{n->oo} (log(n) + gamma - n*exp(gamma) / Ei(log(n) + gamma)) = 1, qed. (End)
EXAMPLE
1/1 + 1/(1 + 1/2) + 1/(1 + 1/2 + 1/3) = 73/33, so a(3) = 73.
MATHEMATICA
f[n_] := Numerator[ Sum[ 1/HarmonicNumber[j], {j, 1, n}]]; Table[ f[n], {n, 0, 14}] (* Robert G. Wilson v, Aug 21 2004 *)
PROG
(PARI) m=13; for(n=0, m, print1(numerator(sum(k=1, n, 1/sum(j=1, k, 1/j))), ", ")) \\ Klaus Brockhaus, Aug 21 2004
CROSSREFS
KEYWORD
easy,frac,nonn
AUTHOR
Leroy Quet, Aug 19 2004
EXTENSIONS
More terms from Klaus Brockhaus and Robert G. Wilson v, Aug 21 2004
STATUS
approved