OFFSET
1,1
COMMENTS
Conjecture: this sequence is infinite. More generally for any fixed integers a,b,c,d >= 1, there are infinitely many k's such that c*d*L(H(k,a)^b) = a*b*L(H(k,c)^d) where L(x) is the number of terms in the continued fraction of x and H(k,r) = Sum_{u=1..k} 1/u^r. Here, (a,b,c,d) = (2,1,1,1).
MATHEMATICA
c[n_, r_] := Length @ ContinuedFraction @ HarmonicNumber[n, r]; Select[Range[10^4], c[#, 2] == 2 * c[#, 1] &] (* Amiram Eldar, Oct 04 2020 *)
PROG
(PARI) H1=H2=1; for(n=2, 10000, H1=H1+1/n; H2=H2+1/n^2; if(length(contfrac(H2))==2*length(contfrac(H1)), print1(n, ", ")))
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Oct 04 2020
STATUS
approved