A226161 Least positive integer k such that 1 + 1/2 + ... + 1/k > n/2.

1, 2, 3, 4, 7, 11, 19, 31, 51, 83, 137, 227, 373, 616, 1015, 1674, 2759, 4550, 7501, 12367, 20390, 33617, 55425, 91380, 150661, 248397, 409538, 675214, 1113239, 1835421, 3026097, 4989191, 8225785, 13562027, 22360003, 36865412, 60780790, 100210581, 165219316

Offset: 1

Author

Clark Kimberling, May 29 2013

Keywords

• nonn

Comments

Conjecture: a(n+1)/a(n) converges to 1.64872...

The conjecture is correct, a(n+1)/a(n) ~ exp(1/2) (A019774). - Charles R Greathouse IV, Jun 03 2013

Conjecture: a(n) = round(exp(n/2-gamma)) for all n, where gamma is the Euler-Mascheroni constant (see A001620). - Jon E. Schoenfield, Jul 19 2015

The terms up to a(52) contained in the b-file have been obtained by working with quadruple-precision (128 bits) floating point numbers, hence there is a very small probability they are off by 1. - Giovanni Resta, Jul 21 2015

All terms in the b-file are correct. Moreover, the above conjecture that a(n) = round(exp(n/2-gamma)) has been verified for all n in 1..10000. - Jon E. Schoenfield, Jul 22 2015

Examples

			a(5) = 7 because 1 + 1/2 + ... + 1/6 < 5/2 < 1 + 1/2 + ... + 1/6 + 1/7.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..52

Crossrefs

Cf. A001620, A019774, A226160.

Programs

Mathematica

nn = 24; g = 1/2; f[n_] := 1/n; a[1] = 1; Do[s = 0; a[n] = NestWhile[# + 1 &, 1, ! (s += f[#]) > n*g &], {n, nn}]; Map[a, Range[nn]]

PARI

first(m)=my(v=vector(m),i,k);for(i=1,m,k=1;while(sum(x=1,k,1/x)<=i/2,k++);v[i]=k;);v; \\ Anders Hellström, Jul 19 2015

Extensions

a(29)-a(35) from Jean-François Alcover, Jun 04 2013

a(36)-a(37) from Jon E. Schoenfield, Aug 31 2013

a(38)-a(39) from Jon E. Schoenfield, Jul 19 2015