A392896 a(n) = least k >= 2 such that Sum_{i=2..k} 1/log(i) >= n.

2, 3, 4, 6, 8, 10, 13, 15, 18, 21, 24, 27, 31, 34, 38, 41, 45, 49, 53, 57, 61, 65, 69, 74, 78, 82, 87, 91, 96, 101, 105, 110, 115, 119, 124, 129, 134, 139, 144, 149, 154, 159, 164, 169, 174, 179, 184, 190, 195, 200, 206, 211, 216, 222, 227, 233, 238, 243, 249, 255, 260, 266, 271, 277

Offset

1,1

Comments

The sum Sum_{i=2..k} 1/log(i) is asymptotic to li(k) - li(2), where li is the logarithmic integral. Hence a(n) is approximately the inverse logarithmic integral of n + li(2).

Analogous to A004080 (least k such that the harmonic number H(k) >= n), but with 1/log(k) replacing 1/k.

The first differences are nondecreasing and grow approximately as log(n).

Links

Table of n, a(n) for n=1..64.

Formula

a(n) ~ n*log(n) as n -> infinity. More precisely, a(n) is approximately li^(-1)(n + li(2)), where li^(-1) is the inverse of the logarithmic integral.

Example

a(1) = 2 because 1/log(2) = 1.4427... >= 1.
a(4) = 6 because 1/log(2) + 1/log(3) + 1/log(4) + 1/log(5) + 1/log(6) = 4.2537... >= 4, while the sum up to k=5 is 3.6956... < 4.

Mathematica

a[n_]:=Module[{k=1}, While[Sum[1/Log[i], {i, 2, k}]<n, k++]; k]; Array[a, 64] (* James C. McMahon, Mar 29 2026 *)

Crossrefs

Cf. A004080.

Sequence in context: A177907 A083006 A352929 * A096061 A100919 A184109

Adjacent sequences: A392893 A392894 A392895 * A392897 A392898 A392899

Keyword

nonn

Author

Firdous Ahmad Mala, Mar 24 2026

Status

approved