OFFSET
1,2
COMMENTS
This sequence gives the number of Fox n-colorings of any 2-bridge knot K with determinant 7. Knots with determinant 7 include the torus knot 7_1 (septafoil) and the twist knot 5_2 (three-twist knot). For a 2-bridge knot K, the double branched cover has H_1(Sigma_2(K)) = Z/det(K)Z; when det(K) = 7 this gives Col_n(K) = n * gcd(7, n) for all n >= 1. More generally, for any 2-bridge knot K with determinant D, Col_n(K) = n * gcd(D, n).
REFERENCES
C. C. Adams, The Knot Book, W. H. Freeman, 1994.
G. Burde and H. Zieschang, Knots, de Gruyter, 2nd ed., 2003.
R. H. Fox, A quick trip through knot theory, in Topology of 3-Manifolds, Prentice-Hall, 1962, pp. 120-167.
FORMULA
a(n) = n * A109010(n).
a(n) = n if 7 does not divide n; a(7k) = 49k.
Multiplicative with a(7^e) = 7^(e+1), a(p^e) = p^e for prime p != 7.
Dirichlet g.f.: zeta(s-1) * (1 + 6/7^(s-1)).
Limit_{N->oo} (1/N)*Sum_{n=1..N} a(n)/n = 13/7.
EXAMPLE
a(7) = 7 * gcd(7,7) = 49.
a(14) = 14 * gcd(7,14) = 98.
For the knot 5_2 (det=7, 5 arcs, 5 crossings): the matrix of Fox coloring relations has Smith Normal Form with invariant factor 7. Exhaustive enumeration over Z_7^5 gives Col_7(5_2) = 49 total 7-colorings (7 trivial, 42 nontrivial).
MATHEMATICA
a[n_] := n * GCD[7, n]; Array[a, 100] (* Amiram Eldar, Mar 13 2026 *)
PROG
(Python)
from math import gcd
def a(n): return n * gcd(7, n)
print([a(n) for n in range(1, 31)])
(PARI) a(n) = n*gcd(n, 7); \\ Michel Marcus, Mar 19 2026
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
Remco Havenaar, Mar 13 2026
STATUS
approved
