A380787 Odd positive integers k whose continued fraction for sqrt(k) has a central term equal to either floor(sqrt(k)) or floor(sqrt(k)) - 1.

3, 7, 11, 19, 23, 27, 31, 43, 47, 51, 59, 67, 71, 79, 83, 103, 107, 119, 123, 127, 131, 139, 151, 163, 167, 171, 179, 187, 191, 199, 211, 223, 227, 239, 243, 251, 263, 267, 271, 283, 287, 291, 307, 311, 331, 339, 343, 347, 359, 363, 367, 379, 383, 387, 391

Offset: 1

Author

Giorgos Kalogeropoulos, Feb 03 2025

Keywords

• nonn

Comments

Conjecture: All terms are congruent to 3 mod 4 and all primes of this form (A002145) are terms of the sequence.

Examples

			71 is a term because the central element of CF(sqrt(71)) = [8; 2, 2, 1, 7, 1, 2, 2, 16] is 7 and floor(sqrt(71)) - 1 = 7.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..8300

Crossrefs

Cf. A002145, A010335, A308778.

Programs

Maple

filter:= proc(n) local L,v;
  if issqr(n) then return false fi;
  L:= map(op, numtheory:-cfrac(sqrt(n),periodic,quotients));
  if nops(L)::even then return false fi;
  v:=L[(1+nops(L))/2]-floor(sqrt(n));
  v = 0 or v = -1
end proc:
select(filter, [seq(i,i=1..500,2); # Robert Israel, Mar 03 2025

Mathematica

Select[2Range@200+1,(l=Last@ContinuedFraction@Sqrt[#]; m=l[[Floor[Length@l/2]]];m==Floor@Sqrt@#||m==Floor@Sqrt@#-1)&]