A031749 - OEIS

%I #38 Jul 12 2021 02:04:10

%S 5043,20168,45375,80664,126035,181488,247023,322640,408339,504120,

%T 609983,725928,851955,988064,1134255,1290528,1456883,1633320,1819839,

%U 2016440,2223123,2439888,2666735,2903664,3150675,3407768,3674943,3952200,4239539

%N Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 71.

%C The continued fraction expansion of sqrt((j*m)^2+t*m) for m >= 1 where t divides 2*j has the form [j*m, 2*j/t, 2*j*m, 2*j/t, 2*j*m, ...]. Thus numbers of the form (71*m)^2 + 2*m for m >= 1 are in the sequence. Are there any others? - _Chai Wah Wu_, Jun 18 2016

%C The term 25776072 is not of the form (71*m)^2 + 2*m. - _Chai Wah Wu_, Jun 19 2016

%H Chai Wah Wu, <a href="/A031749/b031749.txt">Table of n, a(n) for n = 1..10000</a>

%t lt71Q[n_]:=Module[{s=Sqrt[n]},If[IntegerQ[s],0,Min[ContinuedFraction[s] [[2]]]] == 71]; Select[Range[43*10^5],lt71Q] (* _Harvey P. Dale_, Apr 11 2017 *)

%o (Python)

%o from sympy import continued_fraction_periodic

%o A031749_list = [n for n, d in ((n, continued_fraction_periodic(0,1,n)[-1]) for n in range(1,10**5)) if isinstance(d, list) and min(d) == 71] # _Chai Wah Wu_, Jun 09 2017

%Y Cf. A031424.

%K nonn

%O 1,1

%A _David W. Wilson_