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%I #34 Jul 24 2015 23:13:12
%S 1,2,3,4,7,11,19,31,51,83,137,227,373,616,1015,1674,2759,4550,7501,
%T 12367,20390,33617,55425,91380,150661,248397,409538,675214,1113239,
%U 1835421,3026097,4989191,8225785,13562027,22360003,36865412,60780790,100210581,165219316
%N Least positive integer k such that 1 + 1/2 + ... + 1/k > n/2.
%C Conjecture: a(n+1)/a(n) converges to 1.64872...
%C The conjecture is correct, a(n+1)/a(n) ~ exp(1/2) (A019774). - _Charles R Greathouse IV_, Jun 03 2013
%C 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
%C 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
%C 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
%H Giovanni Resta, <a href="/A226161/b226161.txt">Table of n, a(n) for n = 1..52</a>
%e a(5) = 7 because 1 + 1/2 + ... + 1/6 < 5/2 < 1 + 1/2 + ... + 1/6 + 1/7.
%t 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]]
%o (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
%Y Cf. A001620, A019774, A226160.
%K nonn
%O 1,2
%A _Clark Kimberling_, May 29 2013
%E a(29)-a(35) from _Jean-François Alcover_, Jun 04 2013
%E a(36)-a(37) from _Jon E. Schoenfield_, Aug 31 2013
%E a(38)-a(39) from _Jon E. Schoenfield_, Jul 19 2015
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