cp's OEIS Frontend

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

%I A013946 #30 Mar 25 2024 12:05:58
%S A013946 5,2,13,5,29,10,53,17,85,26,5,37,173,2,229,65,293,82,365,101,445,122,
%T A013946 533,145,629,170,733,197,5,226,965,257,1093,290,1229,13,1373,362,61,
%U A013946 401,1685,442,1853,485,2029,530,2213,577,2405,626,2605,677,2813,730,3029,785,3253
%N A013946 Least d for which the number with continued fraction [n,n,n,n...] is in Q(sqrt(d)).
%C A013946 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
%H A013946 Clark Kimberling, <a href="/A013946/b013946.txt">Table of n, a(n) for n = 1..1000</a>
%H A013946 Robin James Spivey, <a href="https://doi.org/10.7546/nntdm.2019.25.3.170-184">Close encounters of the golden and silver ratios</a>, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 3, 170-184.
%H A013946 Ofer Yifrach-Stav, <a href="https://hal.science/tel-04513104">Fast and Private Pool Testing and Contributions to Experimental Mathematics</a>, Doctoral thesis, École normale supérieure (Paris, France), HAL Science [math.cs] 2024, Art. No. tel-04513104. See p. 104.
%F A013946 a(n) = A007913(n^2+4). - _David W. Wilson_, Dec 08 2010
%t A013946 z = 5000; u = Table[{p, e} = Transpose[FactorInteger[n]];
%t A013946 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 *)
%o A013946 (PARI) A013946(n)=core(n^2+4)  \\ _M. F. Hasler_, Dec 08 2010
%Y A013946 a(n) = 2 is equivalent to "n is in the sequence A077444", a(n) = 5 is equivalent to "n is in the sequence A002878".
%K A013946 nonn
%O A013946 1,1
%A A013946 _Clark Kimberling_
%E A013946 More terms from _David W. Wilson_