%I #28 Feb 26 2022 19:18:38

%S 4,11,83,616,1674,4550,12367,33617,91380,248397,1835421,4989191,

%T 13562027,36865412,272400600,740461601,2012783315,5471312310,

%U 40427833596,298723530401,812014744422,2207284924203,6000022499693

%N Numbers k such that if 2 <= j < k then the fractional part of the k-th partial sum of the harmonic series is < the fractional part of the j-th partial sum of the harmonic series.

%C Numbers k such that H(k) sets a new record for being a tiny bit greater than an integer, where H(k) = Sum_{m=1..k} 1/m. For proofs that H(k) is non-integral and almost always a non-terminating decimal see Havil reference.

%C Assuming that H(k) ~= log(k) + gamma + 1/(2k), the next several terms should be 2012783315, 5471312310 and 40427833596; 14872568831 and 109894245429 are not included. - _Robert G. Wilson v_, Aug 14 2003

%D Julian Havil, "Gamma: Exploring Euler's Constant", Princeton University Press, Princeton and Oxford, 2003, pp. 24-25.

%H Steven J. Kifowit, <a href="/A004796/b004796.txt">Table of n, a(n) for n = 1..50</a>

%H T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/harmonic-series">The Harmonic Numbers and Series</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicSeries.html">Harmonic Series</a>.

%e a(2)=11 because H(11) = 3.0198773...; a(3)=83 because H(83) = 5.0020682...

%t s = 0; a = 1; Do[ s = N[s + 1/n, 50]; If[ FractionalPart[s] < a, a = FractionalPart[s]; Print[n]], {n, 2, 1378963718}]

%o (PARI) H(n) = sum(k=1,n,1/k)+0.; { hr(m)=local(rec); rec=0.5; for(n=2,m,if(frac(H(n))<rec, rec=frac(H(n)); print1(n","))) }

%Y Subset of A002387.

%K nonn

%O 1,1

%A _Clark Kimberling_

%E Edited and extended by _Jason Earls_, Jun 30 2003

%E Extended by _Robert G. Wilson v_, Aug 14 2003

%E More terms from _Jon E. Schoenfield_, Mar 26 2010