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Integers n with tan n > |n|, ordered by |n|.
8

%I #47 Mar 07 2021 00:11:19

%S 1,-2,-11,-33,-52174,260515,-573204,37362253,-42781604,122925461,

%T 534483448,3083975227,902209779836,-2685575996367,-65398140378926,

%U 74357078147863,214112296674652,642336890023956,-5920787228742393,-12055686754159438,18190586279576483,-48436859313312404

%N Integers n with tan n > |n|, ordered by |n|.

%C Name was "Positive integers n with |tan n| > n." before signs were added. The sign here shows whether tan(|n|) is positive or negative.

%C That this sequence is infinite was proved by Bellamy, Lagarias and Lazebnik. It seems not to be known whether there are infinitely many n with tan n > n.

%C At approximately 2.37e154, there is a value of n which has tan(n)/n > 556. - _Phil Carmody_, Mar 04 2007 [This is index 214 in the b-file.]

%C As n increases, log(|a(n)|)/n seems to approach Pi/2; this is similar to what would be expected if an integer sequence were created by drawing many random numbers independently from a uniform distribution on the interval [-Pi/2,+Pi/2] and including in the sequence only those integers j for which the j-th random number x_j happened to satisfy |x_j| < 1/j (and applying to j the sign of x_j). - _Jon E. Schoenfield_, Aug 19 2014; updated Nov 07 2014 to reflect the change in the sequence's Name)

%H Jon E. Schoenfield, <a href="/A088306/b088306.txt">Table of n, a(n) for n = 1..1000</a>

%H D. Bellamy, J. C. Lagarias and F. Lazebnik, <a href="http://www.math.udel.edu/~lazebnik/papers/tan_n.pdf">Proposed problem: large values of Tan n</a>

%H Jon E. Schoenfield, <a href="/A088306/a088306_3.txt">Magma program</a>

%p a:=proc(n) if abs(evalf(tan(n)))>n then n else fi end: seq(a(n),n=1..100000); # _Emeric Deutsch_, Dec 18 2004

%t Select[Range[600000],Abs[Tan[#]]>#&] (* _Harvey P. Dale_, Nov 30 2012 *)

%o (PARI) is(n)=tan(n)>abs(n) \\ _Charles R Greathouse IV_, Nov 07 2014

%Y Cf. A000503, A224269, A079330, A088989.

%Y Cf. A249836 (subsequence of positive terms).

%K sign

%O 1,2

%A _Paul Boddington_, Nov 05 2003

%E More terms from _Jon E. Schoenfield_, Aug 17 2014

%E Signs added and other edits by _Franklin T. Adams-Watters_, Sep 09 2014