A060215 Least a(n) such that the period of continued fraction for sqrt(a(n)) has at least n successive 1's.
2, 3, 7, 7, 13, 58, 58, 135, 461, 819, 2081, 13624, 13834, 35955, 95773, 244647, 639389, 1798800, 4374866, 11448871, 30002701, 78439683, 205337953, 541653136, 1407271538
Offset: 0
Examples
n a(n) C.f. period 0 2 1 2 1 3 2 1,2 2 7 4 1,1,1,4 3 7 4 1,1,1,4 4 13 5 1,1,1,1,6 5 58 7 1,1,1,1,1,1,14 6 58 7 1,1,1,1,1,1,14 7 135 8 1,1,1,1,1,1,1,22 8 461 15 2,8,10,1,1,1,1,1,1,1,1,10,8,2,42 9 819 10 1,1,1,1,1,1,1,1,1,56 10 2081 11 1,1,1,1,1,1,1,1,1,1,90 11 13624 62 1,2,1,...,4,5,1,1,1,1,1,1,1,1,1,1,1,5,4,...,1,2,1,232 12 13834 13 1,1,1,1,1,1,1,1,1,1,1,1,234 13 35955 14 1,1,1,1,1,1,1,1,1,1,1,1,1,378 14 95773 25 2,8,2,8,154,1,1,1,1,1,1,1,1,1,1,1,1,1,1,154,8,2,8,2,618 15 244647 16 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,988 16 639389 17 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1598 17 1798800 160 5,5,1,19,...,2,60,(17 1's),60,2,...,19,1,5,5,2682 18 4374866 19 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4182 19 11448871 20 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6766 20 30002701 35 2,8,2,8,2,8,2738,(20 1's),738,8,2,8,2,8,2,10954 21 78439683 22 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,17712 22 205337953 23 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,28658 23 541653136 1442 2,3,1,...,4,581,(23 1's),581,4,...,1,3,2,46546 24 1407271538 25 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,75026
Crossrefs
Cf. A071296.
Programs
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Mathematica
Table[Block[{k = 2}, While[NoneTrue[Map[Length, Select[If[IntegerQ@ #, {{#}}, Split@ Last@ ContinuedFraction@ #] &@ Sqrt@ k, First@ # == 1 &]], # >= n &], k++]; k], {n, 14}] (* Michael De Vlieger, Oct 26 2017 *) -
PARI
print1("2, 3, 7, ");n=3;for(k=1,10^6,v=contfrac(sqrt(k));s=0; for(l=1,length(v)-n,if(v[l]==1,s=s+1,s=0);if(s==n,print1(k", ");n=n+1;k=k-1;break)))
Extensions
More terms from Ralf Stephan, Mar 26 2003
Definition clarified by T. D. Noe, Apr 07 2014
a(11), a(14) corrected and a(17)-a(23) added by Lars Blomberg, Oct 24 2015
a(0), a(24) from Chai Wah Wu, Sep 23 2019
Comments