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).
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
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Jan 15 2013
EXTENSIONS
More terms from Jean-François Alcover, Mar 04 2013
STATUS
approved