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A051764 Number of torus knots with n crossings. 4
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,15

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

D. Bar-Natan, 36 Torus Knots(with up to 36 crossings)

Jim Hoste, Morwen Thistlethwaite, Jeff Weeks, The First 1,701,936 Knots, Math. Intell., 20, 33-48, Fall 1998.

Andrei Malyutin, On the question of genericity of hyperbolic knots, arXiv preprint arXiv:1612.03368 [math.GT], 2016.

Kunio Murasugi, On the braid index of alternating links, Trans. Amer. Math. Soc. 326 (1991), 237-260.

R. G. Scharein, Torus knots and links by crossing number

Eric Weisstein's World of Mathematics, Hyperbolic Knot

Eric Weisstein's World of Mathematics, Knot

Eric Weisstein's World of Mathematics, Torus Knot

FORMULA

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

MAPLE

with(numtheory):

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

seq (a(n), n=1..100);  # Alois P. Heinz, Apr 25 2012

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Sequence in context: A214438 A173432 A101675 * A268533 A275849 A025906

Adjacent sequences:  A051761 A051762 A051763 * A051765 A051766 A051767

KEYWORD

nonn,nice

AUTHOR

Eric W. Weisstein

STATUS

approved

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Last modified March 30 06:59 EDT 2020. Contains 333119 sequences. (Running on oeis4.)