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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, 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).
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 A373422 A328999
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Jan 15 2013
EXTENSIONS
More terms from Jean-François Alcover, Mar 04 2013
STATUS
approved