A226160 Least positive integer k such that 1 + 1/2 + ... + 1/k > n/tau, where tau = golden ratio = (1+sqrt(5))/2.

1, 2, 4, 7, 12, 23, 42, 79, 146, 271, 503, 934, 1732, 3214, 5963, 11063, 20524, 38078, 70646, 131067, 243166, 451140, 836989, 1552846, 2880960, 5344978, 9916415, 18397696, 34132822, 63325839

Offset: 1

Author

Clark Kimberling, May 29 2013

Keywords

• nonn

Comments

Conjecture: a(n+1)/a(n) converges to 1.8552...

Conjecture confirmed: using series expansion of HarmonicNumber(k) one gets a(n+1)/a(n) -> exp(1/tau) = 1.855276958... [Jean-François Alcover, Jun 04 2013]

Examples

			a(4) = 7 because 1 + 1/2 + ... + 1/6 < 4*tau < 1 + 1/2 + ... + 1/7.
		

Crossrefs

Cf. A226161.

Programs

Mathematica

nn = 24; g = 1/GoldenRatio; f[n_] := 1/n; a[1] = 1; Do[s = 0; a[n] = NestWhile[# + 1 &, 1, ! (s += f[#]) > n*g &], {n, 1, nn}]; Map[a,  Range[nn]]

Extensions

More terms from Jean-François Alcover, Jun 04 2013