OFFSET
2,1
COMMENTS
Because lim_{x->oo} (Sum_{k=2..x} 1 / (k*log(log(k)))) - li(log(x)) = 2.7977647035208... (see A363078) then a(n) = round(w) where w is the solution of the equation li(log(w)) + 2.7977647035208... = n.
LINKS
Pascal Sebah, Table of n, a(n) for n = 2..35
FORMULA
For n >= 3, a(n) = round(w) where w is the solution of the equation li(log(w)) + 2.7977647035208... = n.
EXAMPLE
a(2) = 3 because Sum_{k=2..3} 1/(k*log(log(k))) = 2.18008755... > 2 and Sum_{k=2..2} 1/(k*log(log(k))) = -1.364208386450... < 2.
a(7) = 389 because Sum_{k=2..389} 1/(k*log(log(k))) = 7.000345... > 7 and Sum_{k=2..388} 1/(k*log(log(k))) = 6.99890560988... < 7.
MATHEMATICA
(*slow procedure*)
lim = 2; sum = 0; aa = {}; Do[sum = sum + N[1/(k Log[Log[k]]), 100];
If[sum >= lim, AppendTo[aa, k]; Print[{lim, sum, k}];
lim = lim + 1], {k, 2, 269777}]; aa
(*quick procedure *)
aa = {3}; cons = 2.79776470352080492766050456553352884330850083202326989577856315;
Do[ww = w /. NSolve[LogIntegral[Log[w]] + cons == n, w];
AppendTo[aa, Round[ww][[1]]], {n, 3, 21}]; aa
CROSSREFS
KEYWORD
nonn
AUTHOR
Artur Jasinski, Jun 11 2023
STATUS
approved