A289183 -id:A289183 - cp's OEIS frontend

A103762 a(n) = least k with Sum_{j = n..k} 1/j >= 1.

1, 4, 7, 10, 12, 15, 18, 20, 23, 26, 29, 31, 34, 37, 39, 42, 45, 48, 50, 53, 56, 58, 61, 64, 67, 69, 72, 75, 77, 80, 83, 86, 88, 91, 94, 97, 99, 102, 105, 107, 110, 113, 116, 118, 121, 124, 126, 129, 132, 135, 137, 140, 143, 145, 148, 151, 154, 156, 159, 162

Offset: 1

David W. Wilson, Apr 14 2008

nonn

a(n) = A136617(n) + n for n > 1. Also a(n) = A136616(n-1) + 1 for n > 1.
If you compare this to floor(e*n) = A022843, 2,5,8,10,13,16,..., it appears that floor(e*n)-a(n) = 1,1,1,0,1,1,1,1,1,1,0,..., initially consisting of 0's and 1's. The places where the 0's occur are 4, 11, 18, 25, 32, 36, 43, 50, 57, 64, 71, ... whose differences seem to be 4, 7 or 11.
There are some rather sharp estimates on this type of differences between harmonic numbers in Theorem 3.2 of the Sintamarian reference, which may help to uncover such a pattern. - R. J. Mathar, Apr 15 2008
a(n) = round(e*(n-1/2)) with the exception of the terms of A277603; at those values of n, a(n) = round(e*(n-1/2)) + 1. - Jon E. Schoenfield, Apr 03 2018

T. D. Noe, Table of n, a(n) for n = 1..1000
E. R. Bobo, A sequence related to the harmonic series, College Math. J. 26 (1995), 308-310.
D. T. Clancy and S. J. Kifowit, A closer look at Bobo's sequence, College Math. J. 45 (2014), 199-206.
A. Sintamarian, A generalization of Euler's constant, Numer. Algor. 46 (2007), pp. 141-151.

Cf. A136616, A136617, A242679 (Bobo numbers).

Cf. A081881, A277603, A289183. - Jon E. Schoenfield, Mar 31 2018

i = 0; s = 0; Table[While[s < 1, i++; s = s + 1/i]; s = s - 1/n; i, {n, 100}] (* T. D. Noe, Jun 26 2012 *)

default(realprecision, 10^5); e=exp(1);
a(n) = if(n<2, 1, floor(e*n+(1-e)/2+(e-1/e)/(24*n-12))); \\ Jinyuan Wang, Mar 06 2020

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

Original entry on oeis.org

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

Clark Kimberling, May 30 2013

nonn

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

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

Cf. A226184, A289183.

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 *)

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

A226184 Least positive integer k such that 1 + 1/2 + ... + 1/n < 1/(n+1) + ... + 1/(n+k).

Original entry on oeis.org

3, 9, 19, 32, 49, 69, 93, 121, 152, 187, 225, 266, 312, 361, 413, 469, 529, 592, 658, 729, 802, 880, 961, 1045, 1133, 1225, 1320, 1419, 1521, 1627, 1736, 1849, 1966, 2086, 2210, 2337, 2468, 2602, 2740, 2882, 3027, 3175, 3327, 3483, 3642, 3805, 3972, 4142

Offset: 1

Clark Kimberling, May 30 2013

nonn

			a(3) = 19 because 1/4 + 1/5 + ... + 1/(3+18) < 1 + 1/2 + 1/3 < 1/4 + 1/5 + ... + 1/(3+19).

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

Cf. A226183, A289183.

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 *)

a(n) = A226183(n) - n. - Michel Marcus, Sep 09 2021

cp's OEIS Frontend

A103762 a(n) = least k with Sum_{j = n..k} 1/j >= 1.

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

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A226184 Least positive integer k such that 1 + 1/2 + ... + 1/n < 1/(n+1) + ... + 1/(n+k).

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