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%I #31 Mar 06 2020 08:09:35
%S 1,4,7,10,12,15,18,20,23,26,29,31,34,37,39,42,45,48,50,53,56,58,61,64,
%T 67,69,72,75,77,80,83,86,88,91,94,97,99,102,105,107,110,113,116,118,
%U 121,124,126,129,132,135,137,140,143,145,148,151,154,156,159,162
%N a(n) = least k with Sum_{j = n..k} 1/j >= 1.
%C a(n) = A136617(n) + n for n > 1. Also a(n) = A136616(n-1) + 1 for n > 1.
%C If you compare this to floor(e*n) = A022843, 2,5,8,10,13,16,..., it appears that floor(e*n)-a(n) = 1,1,1,0,1,1,1,1,1,1,0,..., initially consisting of 0's and 1's. The places where the 0's occur are 4, 11, 18, 25, 32, 36, 43, 50, 57, 64, 71, ... whose differences seem to be 4, 7 or 11.
%C There are some rather sharp estimates on this type of differences between harmonic numbers in Theorem 3.2 of the Sintamarian reference, which may help to uncover such a pattern. - _R. J. Mathar_, Apr 15 2008
%C a(n) = round(e*(n-1/2)) with the exception of the terms of A277603; at those values of n, a(n) = round(e*(n-1/2)) + 1). - _Jon E. Schoenfield_, Apr 03 2018
%H T. D. Noe, <a href="/A103762/b103762.txt">Table of n, a(n) for n = 1..1000</a>
%H E. R. Bobo, <a href="http://www.jstor.org/stable/2687034">A sequence related to the harmonic series</a>, College Math. J. 26 (1995), 308-310.
%H D. T. Clancy and S. J. Kifowit, <a href="http://dx.doi.org/10.4169/college.math.j.45.3.199">A closer look at Bobo's sequence</a>, College Math. J. 45 (2014), 199-206.
%H A. Sintamarian, <a href="http://dx.doi.org/10.1007/s11075-007-9132-0">A generalization of Euler's constant</a>, Numer. Algor. 46 (2007), pp. 141-151.
%t i = 0; s = 0; Table[While[s < 1, i++; s = s + 1/i]; s = s - 1/n; i, {n, 100}] (* _T. D. Noe_, Jun 26 2012 *)
%o (PARI) default(realprecision, 10^5); e=exp(1);
%o a(n) = if(n<2, 1, floor(e*n+(1-e)/2+(e-1/e)/(24*n-12))); \\ _Jinyuan Wang_, Mar 06 2020
%Y Cf. A136616, A136617, A242679 (Bobo numbers).
%Y Cf. A081881, A277603, A289183. - _Jon E. Schoenfield_, Mar 31 2018
%K nonn
%O 1,2
%A _David W. Wilson_, Apr 14 2008
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