A289183 a(n) is the greatest m such that 2*H(n) > H(m), where H(n) is the n-th harmonic number.

3, 10, 21, 35, 53, 74, 99, 128, 160, 196, 235, 277, 324, 374, 427, 484, 545, 609, 676, 748, 822, 901, 983, 1068, 1157, 1250, 1346, 1446, 1549, 1656, 1766, 1880, 1998, 2119, 2244, 2372, 2504, 2639, 2778, 2921, 3067, 3216, 3369, 3526, 3686, 3850, 4018, 4189

Offset: 1

Author

Joseph Wheat, Jun 27 2017

Keywords

• nonn

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Harmonic Number

Crossrefs

Cf. A000217, A002387, A002805, A092315.

Programs

Mathematica

s = HarmonicNumber@ Range[10^4]; Table[Position[s, k_ /; k < 2 HarmonicNumber@ n][[-1, 1]], {n, 48}] (* Michael De Vlieger, Jun 27 2017 *)
(* The following program searches for such n that f(n) <> a(n) *)
f[n_] := Floor[n^2*E^(EulerGamma + 1/n) - (1/2 + (1/6)*E^(EulerGamma))];
harmonic[n_] := Log[n] + EulerGamma + 1/(2 n) - Sum[BernoulliB[2 k]/(2 k*n^(2 k)), {k, 1, 10}];
Select[Range[100000], 2*harmonic[#] < harmonic[f[#]] &]
(* Vaclav Kotesovec, Jul 17 2017 *)

PARI

a(n) = {my(m=1); hn = sum(k=1, n, 1/k); hm = 1; until(hm > 2*hn, m++; hm+=1/m); m--;} \\ Michel Marcus, Jul 19 2017

Python

from sympy import harmonic
def a(n):
  hn2 = 2 * harmonic(n)
  m = n
  while harmonic(m) <= hn2: m += 1
  return m - 1
print([a(n) for n in range(1, 49)]) # Michael S. Branicky, Mar 10 2021

Formula

From Jon E. Schoenfield, Jul 13 2017: (Start)

It seems that, for the vast majority of values of n > 1, f(n) = floor(n^2 * exp(gamma + 1/n) - C), where gamma is the Euler-Mascheroni constant (A001620) and C = 1/2 + (1/6)*exp(gamma) = 0.7968454029983663308727506838511965915282742..., is equal to a(n); f(n) = a(n) for all n in [2..10000] except n=66: f(66)=7876, but a(66)=7875. [Thanks to Vaclav Kotesovec for identifying the value of C.]

Is there any n > 66 at which f(n) and a(n) differ?

(End)

From Vaclav Kotesovec, Jul 17 2017: (Start)

f(39087) = 2721180603, but a(39087) = 2721180602;

f(517345) = 476697560917, but a(517345) = 476697560916;

f(2013005) = 7217245877275, but a(2013005) = 7217245877274;

No other such numbers below 10000000.

(End)

After 2013005, the only other numbers n < 4*10^9 at which f(n) and a(n) differ are 10240491 and 80968833. - Jon E. Schoenfield, Aug 05 2017

Extensions

More terms from Michael De Vlieger, Jun 27 2017