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 A004796 n-th positive integer k such that if 2 <= j < k then the fractional part of the k-th partial sum of harmonic series is < the fractional part of the j-th partial sum of the harmonic series. 1
 4, 11, 83, 616, 1674, 4550, 12367, 33617, 91380, 248397, 1835421, 4989191, 13562027, 36865412, 272400600, 740461601, 2012783315, 5471312310, 40427833596, 298723530401, 812014744422, 2207284924203, 6000022499693 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS n such that H(n) sets a new record for being a tiny bit greater than an integer, where H(n)=sum_{k=1..n} 1/k. For proofs that H(n) is non-integral and almost always a non-terminating decimal see Havil reference. Assuming that H(n) ~= Ln(n) + gamma + 1/(2n), the next several entries should be 2012783315, 5471312310 and 40427833596; 14872568831 and 109894245429 are not included. - Robert G. Wilson v. REFERENCES Julian Havil, "Gamma: Exploring Euler's Constant", Princeton University Press, Princeton and Oxford, pp. 24-25, 2003. LINKS 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))

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Last modified June 19 19:14 EDT 2013. Contains 226416 sequences.