Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #7 Jun 25 2013 17:31:58
%S 1,2,4,5,7,8,10,11,13,15,16,18,19,21,22,24,26,27,29,30,32,33,35,36,38,
%T 40,41,43,44,46,47,49,51,52,54,55,57,58,60,61,63,65,66,68,69,71,72,74,
%U 76,77,79,80,82,83,85,86,88,90,91,93,94,96,97,99,100,102
%N Position of log n in the joint ranking of harmonic numbers H(k) and {log k}, for k >= 1; complement of A226894.
%C If, in the definition, log k is replaced by g + log k, where g is the Euler-Mascheroni constant, then the position of log n is 2n-1, which leads to a conjecture: limit[1/(H(n) - g - log n) - 2n] = 1/3.
%H Clark Kimberling, <a href="/A226896/b226896.txt">Table of n, a(n) for n = 1..1000</a>
%e log 1 < log 2 < H(1) < log 3 < log 4 < H(2) < ...
%t z = 300; h[n_] := N[HarmonicNumber[n], 100]; t1 = Table[h[n], {n, 1, z}]; t2 = Table[N[Log[n], 100], {n, 1, 3 z}]; t3 = Union[t1, t2]; p[n_] := Position[t3, h[n]]; Flatten[Table[p[n], {n, 1, 3 z}]] (* A226894 *)
%t Differences[%] (* A226895 *)
%t Complement[Range[z], %%] (* A226896 *)
%Y Cf. A001008(n)/A002805(n) (H(n)), A226894.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jun 21 2013