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A215000 a(n) = floor(exp(1 + 1/2 + 1/3 + ... + 1/n)).

%I A215000 #31 May 17 2025 00:34:27
%S A215000 2,4,6,8,9,11,13,15,16,18,20,22,24,25,27,29,31,32,34,36,38,40,41,43,
%T A215000 45,47,48,50,52,54,56,57,59,61,63,65,66,68,70,72,73,75,77,79,81,82,84,
%U A215000 86,88,89,91,93,95,97,98,100,102,104,105,107,109,111,113,114
%N A215000 a(n) = floor(exp(1 + 1/2 + 1/3 + ... + 1/n)).
%C A215000 a(n) is the greatest integer k for which log k < 1 + 1/2 + ... + 1/n.
%C A215000 a(n) is asymptotically equals to n*e^(gamma) for large values of n where 'gamma' is the Euler-Mascheroni constant (Cf. A001620). - _Balarka Sen_, Aug 19 2012
%H A215000 Clark Kimberling, <a href="/A215000/b215000.txt">Table of n, a(n) for n = 1..10000</a>
%e A215000 log 2 < 1 < log 3, so a(1) = 2;
%e A215000 log 4 < 1 + 1 + 1/2 < log 5, so a(2) = 4;
%e A215000 log 6 < 1 + 1/2 + 1/3 < log 7, so a(3) = 6.
%t A215000 f[n_] := Sum[1/h, {h, n}]; Table[Floor[E^f[n]], {n, 100}]
%t A215000 Table[Floor[Exp[HarmonicNumber[n]]], {n, 1, 100}] (* _G. C. Greubel_, Aug 30 2018 *)
%o A215000 (PARI) a(n) = floor(exp(sum(X=1,n,1/X))) \\ _Balarka Sen_, Aug 19 2012
%o A215000 (Magma) [Floor(Exp((&+[1/k :k in [1..n]]))): n in [1..30]]; // _G. C. Greubel_, Feb 01 2018
%Y A215000 Cf. A215001, A001620, A073004.
%K A215000 nonn
%O A215000 1,1
%A A215000 _Clark Kimberling_, Aug 18 2012