cp's OEIS Frontend

A004796 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.

%I A004796 #29 Feb 16 2025 08:32:28
%S A004796 4,11,83,616,1674,4550,12367,33617,91380,248397,1835421,4989191,
%T A004796 13562027,36865412,272400600,740461601,2012783315,5471312310,
%U A004796 40427833596,298723530401,812014744422,2207284924203,6000022499693
%N A004796 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 A004796 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 A004796 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 A004796 Julian Havil, "Gamma: Exploring Euler's Constant", Princeton University Press, Princeton and Oxford, 2003, pp. 24-25.
%H A004796 Steven J. Kifowit, <a href="/A004796/b004796.txt">Table of n, a(n) for n = 1..50</a>
%H A004796 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/harmonic-series">The Harmonic Numbers and Series</a>
%H A004796 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicSeries.html">Harmonic Series</a>.
%e A004796 a(2)=11 because H(11) = 3.0198773...; a(3)=83 because H(83) = 5.0020682...
%t A004796 s = 0; a = 1; Do[ s = N[s + 1/n, 50]; If[ FractionalPart[s] < a, a = FractionalPart[s]; Print[n]], {n, 2, 1378963718}]
%o A004796 (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 A004796 Subset of A002387.
%K A004796 nonn
%O A004796 1,1
%A A004796 _Clark Kimberling_
%E A004796 Edited and extended by _Jason Earls_, Jun 30 2003
%E A004796 Extended by _Robert G. Wilson v_, Aug 14 2003
%E A004796 More terms from _Jon E. Schoenfield_, Mar 26 2010