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A051764 Number of torus knots with n crossings. 4


%S 0,0,1,0,1,0,1,1,1,1,1,0,1,1,2,1,1,0,1,1,2,1,1,1,1,1,2,2,1,0,1,2,2,1,

%T 2,1,1,1,2,1,1,0,1,2,2,1,1,2,1,1,2,2,1,1,2,2,2,1,1,1,1,1,3,2,2,1,1,2,

%U 2,1,1,2,1,1,2,2,2,1,1,2,2,1,1,1,2,1,2,3,1,1,2,2,2,1,2,2,1,1,3,1,1,1,1,3,3

%N Number of torus knots with n crossings.

%H Alois P. Heinz, <a href="/A051764/b051764.txt">Table of n, a(n) for n = 1..10000</a>

%H D. Bar-Natan, <a href="http://katlas.org/wiki/36_Torus_Knots">36 Torus Knots(with up to 36 crossings)</a>

%H Jim Hoste, Morwen Thistlethwaite, Jeff Weeks, <a href="http://dx.doi.org/10.1007/BF03025227">The First 1,701,936 Knots</a>, Math. Intell., 20, 33-48, Fall 1998.

%H Andrei Malyutin, <a href="https://arxiv.org/abs/1612.03368">On the question of genericity of hyperbolic knots</a>, arXiv preprint arXiv:1612.03368 [math.GT], 2016.

%H Kunio Murasugi, <a href="https://doi.org/10.1090/S0002-9947-1991-1000333-3">On the braid index of alternating links</a>, Trans. Amer. Math. Soc. 326 (1991), 237-260.

%H R. G. Scharein, <a href="http://www.cs.ubc.ca/nest/imager/contributions/scharein/knot-theory/torus_xing.html">Torus knots and links by crossing number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HyperbolicKnot.html">Hyperbolic Knot</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Knot.html">Knot</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TorusKnot.html">Torus Knot</a>

%F a(n) = cardinality of the set {k| sqrt(n) < k <= n and gcd(k, 1+n/k) = 1}; see Murasugi article. - _Hermann Gruber_, Mar 05 2003

%p with(numtheory):

%p a:= n-> nops (select (k-> is (sqrt(n)<k and igcd(k, 1+n/k)=1), divisors(n))):

%p seq (a(n), n=1..100); # _Alois P. Heinz_, Apr 25 2012

%t a[n_] := (r = Reduce[Sqrt[n] < k <= n && GCD[k, 1 + n/k] == 1, k, Integers]; Which[r === False, 0, r[[0]] === Equal, 1, True, Length[r]]); Table[a[n], {n, 1, 105}] (* _Jean-Fran├žois Alcover_, Jan 16 2013 *)

%o (PARI) a(n)=my(t=sqrtint(n));sumdiv(n,k,k>t && gcd(k,n/k+1)==1) \\ _Charles R Greathouse IV_, Apr 26 2012

%K nonn,nice

%O 1,15

%A _Eric W. Weisstein_

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Last modified January 18 16:27 EST 2020. Contains 331011 sequences. (Running on oeis4.)