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Least positive integer k such that 1 + 1/2 + ... + 1/k > 2n/3.
1

%I #10 Jan 04 2019 21:39:23

%S 1,2,4,8,16,31,60,116,227,441,859,1674,3260,6349,12367,24088,46916,

%T 91380,177984,346666,675214,1315136,2561536,4989191,9717617,18927334,

%U 36865412

%N Least positive integer k such that 1 + 1/2 + ... + 1/k > 2n/3.

%C Conjecture: a(n+1)/a(n) converges to 1.94...

%C Conjecture confirmed: series expansion of HarmonicNumber(k) gives a(n+1)/a(n) -> exp(2/3) = 1.947734... [_Jean-François Alcover_, Jun 05 2013]

%e a(8) = 116 because 1 + 1/2 + ... + 1/115 < 16/3 < 1 + 1/2 + ... + 1/116.

%t z = 18; f[n_] := 1/n; Do[s = 0; a[n] = NestWhile[# + 1 &, 1, ! (s += f[#]) >= 2n/3 &], {n, 1, z}]; m = Map[a, Range[z]]

%Y Cf. A226186, A226187.

%K nonn,more

%O 1,2

%A _Clark Kimberling_, May 30 2013

%E More terms from _Jean-François Alcover_, Jun 05 2013

%E Deleted obsolete b-file. - _N. J. A. Sloane_, Jan 04 2019