A226186 -id:A226186 - cp's OEIS frontend

A226187 Least positive integer k such that 1 + 1/2 + ... + 1/k > n/3.

1, 1, 1, 2, 3, 4, 6, 8, 11, 16, 22, 31, 43, 60, 83, 116, 162, 227, 316, 441, 616, 859, 1199, 1674, 2336, 3260, 4550, 6349, 8861, 12367, 17259, 24088, 33617, 46916, 65477, 91380, 127531, 177984, 248397, 346666, 483812, 675214, 942336, 1315136, 1835421, 2561536, 3574912, 4989191, 6962977, 9717617

Offset: 1

Clark Kimberling, May 30 2013

nonn

Conjecture: a(n+1)/a(n) converges to 1.39...

This constant is probably exp(1/3) = 1.395612425086089528628..., see A004080. - Ralf Stephan, Jun 03 2013

			a(10) = 16 because 1 + 1/2 + ... + 1/15 < 10/3 < 1 + 1/2 + ... + 1/16.

Cf. A226186, A226188, A002387, A004080.

z = 32; f[n_] := 1/n; Do[s = 0; a[n] = NestWhile[# + 1 &, 1, ! (s += f[#]) >= n/3 &], {n, 1, z}]; m = Map[a, Range[z]]

a(n)=local(s,k);s=0;k=1;while(s<=n/3,s=s+1/k;k++);k-1

More terms from Jean-François Alcover, Jun 05 2013

Deleted obsolete b-file. - N. J. A. Sloane, Jan 04 2019

A226188 Least positive integer k such that 1 + 1/2 + ... + 1/k > 2n/3.

Original entry on oeis.org

1, 2, 4, 8, 16, 31, 60, 116, 227, 441, 859, 1674, 3260, 6349, 12367, 24088, 46916, 91380, 177984, 346666, 675214, 1315136, 2561536, 4989191, 9717617, 18927334, 36865412

Offset: 1

Clark Kimberling, May 30 2013

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

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

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

Cf. A226186, A226187.

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]]

More terms from Jean-François Alcover, Jun 05 2013

Deleted obsolete b-file. - N. J. A. Sloane, Jan 04 2019

cp's OEIS Frontend

A226187 Least positive integer k such that 1 + 1/2 + ... + 1/k > n/3.

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