A058027 Sum of terms of continued fraction for n-th harmonic number, 1 + 1/2 + 1/3 + ... + 1/n.

1, 3, 7, 14, 15, 10, 16, 19, 26, 35, 72, 41, 38, 79, 83, 42, 59, 143, 68, 61, 70, 51, 50, 78, 74, 82, 130, 113, 111, 315, 235, 1190, 211, 407, 112, 122, 142, 246, 693, 133, 138, 162, 1904, 243, 170, 539, 363, 210, 197, 518, 275, 502, 527, 316, 1729, 224, 228, 909

Offset: 1

Author

Leroy Quet, Nov 15 2000

Keywords

• easy
• nonn

Comments

Is anything known about the asymptotics of this sequence?

Should be asymptotic to D*n^(3/2) with D=0.4.... - Benoit Cloitre, Dec 23 2003

Examples

			1 + 1/2 +1/3 = 11/6 = 1 + 1/(1 + 1/5). So sum of terms of continued fraction is 1 + 1 + 5 = 7.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Harmonic Number
• Eric Weisstein's World of Mathematics, Continued Fraction

Crossrefs

m-th harmonic number H(m) = A001008(m)/A002805(m).

Cf. A055573, A100398, A110020, A112286, A112287.

Programs

Mathematica

Table[Plus @@ ContinuedFraction[HarmonicNumber[n]], {n, 60}] (* Ray Chandler, Sep 17 2005 *)

PARI

a(n) = vecsum(contfrac(sum(k=1, n, 1/k))); \\ Michel Marcus, Mar 23 2017