A308778 -id:A308778 - cp's OEIS frontend

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

Giorgos Kalogeropoulos, Feb 03 2025

nonn

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

			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.

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

Cf. A002145, A010335, A308778.

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

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

A308780 First element of the periodic part of the continued fraction expansion of sqrt(k), where the period is 2.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 2, 1, 5, 2, 1, 6, 4, 3, 2, 1, 7, 2, 1, 8, 4, 2, 1, 9, 6, 3, 2, 1, 10, 5, 4, 2, 1, 11, 2, 1, 12, 8, 6, 4, 3, 2, 1, 13, 2, 1, 14, 7, 4, 2, 1, 15, 10, 6, 5, 3, 2, 1, 16, 8, 4, 2, 1, 17, 2, 1, 18, 12, 9, 6, 4, 3, 2, 1, 19, 2, 1

Offset: 1

Georg Fischer, Jun 24 2019

nonn

			The continued fractions for sqrt(3..8) are:
   3 1;1,2
   4 2 (square)
   5 2;4
   6 2;2,4
   7 2;1,1,1,4
   8 2;1,4
Those for 3, 6 and 8 have a period of 2, therefore the sequence starts with 1, 2, 1.

Georg Fischer, Table of the continued fractions of sqrt(0..20000).

Cf. A013642, A308778.

s := proc(n) if not issqr(n) then numtheory[cfrac](sqrt(n), 'periodic', 'quotients')[2]; if nops(%) = 2 then return %[1] fi fi; NULL end:
seq(s(n), n=1..399); # Peter Luschny, Jul 01 2019

Reap[For[k = 3, k <= 399, k++, If[!IntegerQ[Sqrt[k]], cf = ContinuedFraction[Sqrt[k]]; If[Length[cf[[2]]] == 2, Sow[cf[[2, 1]]]]]]][[2, 1]] (* Jean-François Alcover, May 03 2024 *)
(* Second program (much simpler): *)
Table[2 a/b, {a, 1, 20}, {b, Rest@Divisors[2 a]}] // Flatten (* Jean-François Alcover, May 04 2024, after a remark by Kevin Ryde *)

cp's OEIS Frontend

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.

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