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

Author

Clark Kimberling

Keywords

• nonn

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.

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".

Programs

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 *)

PARI

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

Formula

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

Extensions

More terms from David W. Wilson