A070985 -id:A070985 - cp's OEIS frontend

A336088 k such that L(H(k,2)) = 2*L(H(k,1)) 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.

28, 61, 90, 105, 121, 321, 339, 382, 408, 466, 602, 1079, 1121, 1596, 1782, 2067, 2104, 2170, 2220, 2250, 2435, 2456, 2884, 3141, 3242, 3321, 3328, 3435, 4195, 4323, 4348, 4497, 4766, 4914, 5241, 5526, 6290, 6581, 6597, 9306, 9734

Offset: 1

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Benoit Cloitre, Oct 04 2020

nonn

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).

Cf. A055573, A070985.

c[n_, r_] := Length @ ContinuedFraction @ HarmonicNumber[n, r]; Select[Range[10^4], c[#, 2] == 2 * c[#, 1] &] (* Amiram Eldar, Oct 04 2020 *)

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,",")))

cp's OEIS Frontend

A336088 k such that L(H(k,2)) = 2*L(H(k,1)) 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.

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