A013946 Least d for which the number with continued fraction [n,n,n,n...] is in Q(sqrt(d)).

5, 2, 13, 5, 29, 10, 53, 17, 85, 26, 5, 37, 173, 2, 229, 65, 293, 82, 365, 101, 445, 122, 533, 145, 629, 170, 733, 197, 5, 226, 965, 257, 1093, 290, 1229, 13, 1373, 362, 61, 401, 1685, 442, 1853, 485, 2029, 530, 2213, 577, 2405, 626, 2605, 677, 2813, 730, 3029, 785, 3253

Offset

1,1

Comments

Square roots of a(n) are found in the limiting ratios of A000045, A001333, A003688, A015448, A015449, A015451 and so on. I.e., the limiting ratios are the golden ratio, silver mean, bronze ratio and so on. - Mats Granvik, Oct 20 2010

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Robin James Spivey, Close encounters of the golden and silver ratios, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 3, 170-184.

Ofer Yifrach-Stav, Fast and Private Pool Testing and Contributions to Experimental Mathematics, Doctoral thesis, École normale supérieure (Paris, France), HAL Science [math.cs] 2024, Art. No. tel-04513104. See p. 104.

Formula

a(n) = A007913(n^2+4). - David W. Wilson, Dec 08 2010

Mathematica

z = 5000; u = Table[{p, e} = Transpose[FactorInteger[n]];
Times @@ (p^Mod[e, 2]), {n, z}]; Table[u[[n^2 + 4]], {n, 1, Sqrt[z - 4]}]  (* Clark Kimberling, Jul 20 2015, based on T. D. Noe's program at A007913 *)

Prog

(PARI) A013946(n)=core(n^2+4)  \\ M. F. Hasler, Dec 08 2010

Crossrefs

a(n) = 2 is equivalent to "n is in the sequence A077444", a(n) = 5 is equivalent to "n is in the sequence A002878".

Sequence in context: A082153 A283944 A294684 * A261327 A330613 A085436

Adjacent sequences: A013943 A013944 A013945 * A013947 A013948 A013949

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from David W. Wilson

Status

approved