A206582 The least nonsquare number s having exactly n twos in the periodic part of the continued fraction of sqrt(s).

5, 2, 19, 45, 71, 153, 199, 589, 301, 989, 526, 1711, 739, 1633, 631, 3886, 1324, 4897, 2524, 7021, 2374, 4189, 2311, 10033, 3571, 3901, 2326, 8869, 4789, 10873, 6301, 10921, 6451, 11929, 6841, 12709, 7996, 13561, 7351, 19177, 9949, 16969, 12286, 22969, 11341

Offset: 0

Author

T. D. Noe, Mar 19 2012

Keywords

• nonn

Crossrefs

Cf. A206578 (n ones), A206583 (n threes), A206584 (n fours), A206585 (n fives).

Programs

Maple

V:= Array(0..50):  count:= 0:
with(NumberTheory):
for i from 2 while count < 51 do
  if issqr(i) then next fi;
  cf:= Term(ContinuedFraction(sqrt(i)),periodic);
  v:= numboccur(cf[2],2);
  if v <= 50 and V[v] = 0 then
    V[v]:= i; count:= count+1;
  fi;
od:
convert(V,list); # Robert Israel, May 13 2024

Mathematica

nn = 50; zeros = nn; t = Table[0, {nn}]; k = 2; While[zeros > 0, If[! IntegerQ[Sqrt[k]], cnt = Count[ContinuedFraction[Sqrt[k]][[2]], 2]; If[cnt <= nn && t[[cnt]] == 0, t[[cnt]] = k; zeros--]]; k++]; Join[{5}, t]

Extensions

Corrected by Robert Israel, May 13 2024