login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 8 14:38 EST 2019. Contains 329865 sequences. (Running on oeis4.)