|
| |
|
|
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-François 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
|
| |
|
|
