cp's OEIS Frontend

A082912 Least k such that H(k) > 10^n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.

%I A082912 #25 Feb 16 2025 08:32:49
%S A082912 2,12367,15092688622113788323693563264538101449859497
%N A082912 Least k such that H(k) > 10^n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.
%C A082912 "In 1968 John W. Wrench Jr calculated the exact minimum number of terms needed for the series to sum past 100; that number is 15 092 688 622 113 788 323 693 563 264 538 101 449 859 497. Certainly, he did not add up the terms.
%C A082912 "Imagine a computer doing so and suppose that it takes it 10^-9 seconds to add each new term to the sum and that we set it adding and let it continue doing so indefinitely. The job will have been completed in not less than 3.5 * 10^17 (American) billion years." Havil.
%D A082912 Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 23.
%H A082912 R. Baillie, <a href="http://arxiv.org/abs/1105.3943">Fun With Very Large Numbers</a>, arXiv preprint arXiv:1105.3943, 2011
%H A082912 R. P. Boas, Jr. and J. W. Wrench, Jr., <a href="http://www.jstor.org/stable/2316476">Partial sums of the harmonic series</a>, Amer. Math. Monthly, 78 (1971), 864-870.
%H A082912 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%H A082912 Lin Zhang, <a href="http://www.d.umn.edu/math/Technical%20Reports/Technical%20Reports%202007-/TR%202013/Lin%20Zhang%20thesis_masters-4.pdf">A Likelihood Ratio Test of Independence of Components for High-dimensional Normal Vectors</a>, MS Thesis, Univ. Minnesota, 2013.
%F A082912 H(k) ~= log(k) + Euler's Gamma Constant (A001620) + 1/(2k).
%F A082912 a(n) = A002387(10^n). - _Joerg Arndt_, Jul 13 2015
%t A082912 f[n_] := Floor[Exp[n - EulerGamma] - 1/2] + 1; Table[ f[10^n], {n, 0, 2}]
%Y A082912 Cf. A002387, A001620.
%K A082912 nonn,bref
%O A082912 0,1
%A A082912 _Robert G. Wilson v_, Apr 14 2003