A361089 a(n) = smallest integer x such that Sum_{k = 2..x} 1/(k*log(log(k))) > n.

3, 5, 8, 21, 76, 389, 2679, 23969, 269777, 3717613, 61326301, 1188642478, 26651213526, 682263659097, 19720607003199, 637490095320530, 22857266906194526, 902495758030572213, 38993221443197045348, 1833273720522384358862

Offset: 2

Author

Artur Jasinski, Jun 11 2023

Keywords

• nonn

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.

Examples

			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.
		

Links

• Pascal Sebah, Table of n, a(n) for n = 2..35

Crossrefs

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363368, A363078.

Programs

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

Formula

For n >= 3, a(n) = round(w) where w is the solution of the equation li(log(w)) + 2.7977647035208... = n.