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.

4, 11, 83, 616, 1674, 4550, 12367, 33617, 91380, 248397, 1835421, 4989191, 13562027, 36865412, 272400600, 740461601, 2012783315, 5471312310, 40427833596, 298723530401, 812014744422, 2207284924203, 6000022499693

Offset

1,1

Comments

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.

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

References

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

Links

Steven J. Kifowit, Table of n, a(n) for n = 1..50

T. Sillke, The Harmonic Numbers and Series

Eric Weisstein's World of Mathematics, Harmonic Series.

Example

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

Mathematica

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

Prog

(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", "))) }

Crossrefs

Subset of A002387.

Sequence in context: A213328 A181275 A000850 * A320289 A298858 A181267

Adjacent sequences: A004793 A004794 A004795 * A004797 A004798 A004799

Keyword

nonn

Author

Clark Kimberling

Extensions

Edited and extended by Jason Earls, Jun 30 2003

Extended by Robert G. Wilson v, Aug 14 2003

More terms from Jon E. Schoenfield, Mar 26 2010

Status

approved