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

Author

Clark Kimberling

Keywords

• nonn

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

Examples

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

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.

Crossrefs

Subset of A002387.

Programs

Mathematica

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

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))
				

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