

A139036


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


1



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
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OFFSET

1,3


LINKS

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


EXAMPLE

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.


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A100398.
Sequence in context: A163169 A097974 A333753 * A292129 A291970 A281981
Adjacent sequences: A139033 A139034 A139035 * A139037 A139038 A139039


KEYWORD

nonn


AUTHOR

Leroy Quet, May 31 2008


EXTENSIONS

Terms a(10a(15) added by Gonzalo Ciruelos, Aug 02 2013
Corrected and extended by Harvey P. Dale, Nov 24 2016


STATUS

approved



