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

1,1

Links

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Harmonic Number

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

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 *)

Prog

(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

Crossrefs

Cf. A000217, A002387, A002805, A092315.

Sequence in context: A210990 A004194 A097590 * A194141 A281153 A014105

Adjacent sequences: A289180 A289181 A289182 * A289184 A289185 A289186

Keyword

nonn

Author

Joseph Wheat, Jun 27 2017

Extensions

More terms from Michael De Vlieger, Jun 27 2017

Status

approved