A209272 a(1) = 1 and, for n >= 2, a(n) is the least integer such that both p(n) and q(n) are squarefree where p(n)/q(n) is the n-th convergent of the continued fraction [a(1),a(2),...,a(n)].

1, 1, 1, 1, 4, 2, 4, 2, 3, 1, 4, 3, 4, 1, 1, 4, 2, 1, 3, 1, 4, 2, 1, 2, 1, 4, 4, 2, 1, 2, 4, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 4, 2, 4, 2, 3, 4, 1, 2, 4, 1, 1, 3, 1, 1, 3, 1, 1, 4, 3, 1, 3, 1, 1, 4, 2, 1, 2, 1, 3, 1, 3, 2, 3, 1, 3, 1, 2, 4, 2, 3, 1, 3, 1, 2, 1, 6, 4, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 4, 2, 1, 4, 3, 5

Offset

1,5

Comments

Conjecture 1: sequence is unbounded.

Conjecture 2: (a(1)a(2)...a(n))^(1/n) seems to converge to 1.(7)... a limit different from Khintchine's constant (see A002210).

Links

Table of n, a(n) for n=1..109.

Mathematica

Clear[a]; a[1] = 1; a[n_] := a[n] = Catch[ For[k = 1, True, k++, cv = Convergents[ Append[ Table[a[j], {j, 1, n - 1}], k], n] // Last; If[ SquareFreeQ[cv // Numerator] && SquareFreeQ[cv // Denominator], Throw[k]]]]; Table[a[n], {n, 1, 109}] (* Jean-François Alcover, Mar 04 2013 *)

Prog

(PARI)  v=[1]; for(k=1, 100, m=1; while(issquarefree(contfracpnqn(concat(v, [m]))[1, 1])+issquarefree(contfracpnqn(concat(v, [m]))[2, 1])<2, m++); v=concat(v, [m])); a(n)=v[n];

Crossrefs

Sequence in context: A105256 A064127 A178253 * A105397 A328999 A236185

Adjacent sequences: A209269 A209270 A209271 * A209273 A209274 A209275

Keyword

nonn

Author

Benoit Cloitre, Jan 15 2013

Extensions

More terms from Jean-François Alcover, Mar 04 2013

Status

approved