A139036 a(n) = the number of 1's in the continued fraction expansion of the n-th harmonic number, H(n) = Sum_{k=1 to n} 1/k.

1, 1, 2, 0, 2, 0, 2, 3, 5, 3, 1, 4, 6, 2, 3, 8, 8, 5, 8, 4, 10, 8, 8, 8, 7, 12, 9, 10, 13, 9, 8, 5, 10, 9, 12, 17, 15, 7, 9, 13, 8, 14, 12, 13, 14, 12, 11, 18, 17, 21, 19, 11, 12, 18, 16, 21, 33, 28, 19, 14, 20, 31, 19, 17, 21, 21, 16, 28, 23, 19, 18, 27, 40

Offset: 1

Author

Leroy Quet, May 31 2008

Keywords

• nonn

Examples

			The 7th harmonic number is 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 = 363/140, which has the continued fraction representation 2 + 1/(1 + 1/(1 + 1/(2 + 1/(5 + 1/5)))) = [2;1,1,2,5,5]. There are exactly two 1's in the continued fraction representation, so a(7) = 2.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..1000
• Gonzalo Ciruelos, Python script that generates a(1)..a(n)

Crossrefs

Cf. A100398.

Programs

Mathematica

Table[Count[ContinuedFraction[HarmonicNumber[n]],1],{n,100}] (* Harvey P. Dale, Nov 24 2016 *)

PARI

a(n) = #select(x->x==1, contfrac(sum(i=1, n, 1/i))); \\ Jinyuan Wang, Mar 01 2020

Extensions

a(10)-a(15) from Gonzalo Ciruelos, Aug 02 2013

Corrected and extended by Harvey P. Dale, Nov 24 2016