

A226161


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


3



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
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OFFSET

1,2


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/2gamma)) for all n, where gamma is the EulerMascheroni constant (see A001620).  Jon E. Schoenfield, Jul 19 2015
The terms up to a(52) contained in the bfile have been obtained by working with quadrupleprecision (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 bfile are correct. Moreover, the above conjecture that a(n) = round(exp(n/2gamma)) has been verified for all n in 1..10000.  Jon E. Schoenfield, Jul 22 2015


LINKS

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


EXAMPLE

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


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]]


PROG

(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


CROSSREFS

Cf. A001620, A019774, A226160.
Sequence in context: A080074 A317767 A018064 * A188624 A327010 A341823
Adjacent sequences: A226158 A226159 A226160 * A226162 A226163 A226164


KEYWORD

nonn


AUTHOR

Clark Kimberling, May 29 2013


EXTENSIONS

a(29)a(35) from JeanFrançois 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


STATUS

approved



