A226183 Least positive integer k such that 1 + 1/2 + ... + 1/n < 1/(n+1) + ... + 1/k.

4, 11, 22, 36, 54, 75, 100, 129, 161, 197, 236, 278, 325, 375, 428, 485, 546, 610, 677, 749, 823, 902, 984, 1069, 1158, 1251, 1347, 1447, 1550, 1657, 1767, 1881, 1999, 2120, 2245, 2373, 2505, 2640, 2779, 2922, 3068, 3217, 3370, 3527, 3687, 3851, 4019, 4190

Offset: 1

Author

Clark Kimberling, May 30 2013

Keywords

• nonn

Examples

			a(3) = 22 because 1/4 + 1/5 + ... + 1/21 < 1 + 1/2 + 1/3 < 1/4 + 1/5 + ... + 1/22.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A226184, A289183.

Programs

Mathematica

z = 55; f[n_] := 1/n; p[n_] := p[n] = Sum[f[k], {k, 1, n}]; Do[s = 0; a[n] = NestWhile[# + 1 &, 1, ! (s += f[#]) >= 2 p[n] &], {n, 1, z}]; m = Map[a, Range[z]]  (* A226183 *)
m1 = Table[m[[n]] - n, {n, 1, z}] (* A226184 *)

Formula

a(n) = round(exp(gamma)*n*(n+1) + gamma) where gamma is the Euler-Mascheroni constant 0.57714... (A001620). - Carl R. White, Sep 01 2021

a(n) = A226184(n) + n. - Michel Marcus, Sep 09 2021