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

2, 12367, 15092688622113788323693563264538101449859497

Offset: 0

Author

Robert G. Wilson v, Apr 14 2003

Keywords

• nonn
• bref

Comments

"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.

"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.

References

• Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 23.

Links

• R. Baillie, Fun With Very Large Numbers, arXiv preprint arXiv:1105.3943, 2011
• R. P. Boas, Jr. and J. W. Wrench, Jr., Partial sums of the harmonic series, Amer. Math. Monthly, 78 (1971), 864-870.
• Eric Weisstein's World of Mathematics, Harmonic Number
• Lin Zhang, A Likelihood Ratio Test of Independence of Components for High-dimensional Normal Vectors, MS Thesis, Univ. Minnesota, 2013.

Crossrefs

Cf. A002387, A001620.

Programs

Mathematica

f[n_] := Floor[Exp[n - EulerGamma] - 1/2] + 1; Table[ f[10^n], {n, 0, 2}]

Formula

H(k) ~= log(k) + Euler's Gamma Constant (A001620) + 1/(2k).

a(n) = A002387(10^n). - Joerg Arndt, Jul 13 2015