A394222 - OEIS

%I #25 Mar 19 2026 10:12:23

%S 1,2,3,4,25,6,7,8,9,50,11,12,13,14,75,16,17,18,19,100,21,22,23,24,125,

%T 26,27,28,29,150,31,32,33,34,175,36,37,38,39,200,41,42,43,44,225,46,

%U 47,48,49,250,51,52,53,54,275,56,57,58,59,300,61,62,63,64,325,66,67,68,69,350

%N a(n) = n * gcd(5, n).

%C This sequence gives the number of Fox n-colorings of any 2-bridge knot K with determinant 5. Knots with determinant 5 include the torus knot 5_1 (cinquefoil) and the figure-eight knot 4_1. For a 2-bridge knot, the double branched cover is a lens space with H_1(Sigma_2(K)) = Z/det(K)Z; when det(K) = 5 this gives Col_n(K) = n * gcd(5, n) for all n >= 1. More generally, for any 2-bridge knot K with determinant D, Col_n(K) = n * gcd(D, n).

%D C. C. Adams, The Knot Book, W. H. Freeman, 1994.

%D G. Burde and H. Zieschang, Knots, de Gruyter, 2nd ed., 2003.

%D R. H. Fox, A quick trip through knot theory, in Topology of 3-Manifolds, Prentice-Hall, 1962, pp. 120-167.

%F a(n) = n * A109009(n).

%F a(n) = n if 5 does not divide n; a(5k) = 25k.

%F Multiplicative with a(5^e) = 5^(e+1), a(p^e) = p^e for prime p != 5.

%F Dirichlet g.f.: zeta(s-1) * (1 + 4/5^(s-1)).

%F Limit_{N->oo} (1/N)*Sum_{n=1..N} a(n)/n = 9/5.

%e a(5) = 5 * gcd(5,5) = 25.

%e a(10) = 10 * gcd(5,10) = 50.

%e For the knot 4_1 (figure-eight knot, det=5, 4 arcs, 4 crossings): the matrix of Fox coloring relations has Smith Normal Form with invariant factor 5. Exhaustive enumeration over Z_5^4 gives Col_5(4_1) = 25 total 5-colorings (5 trivial, 20 nontrivial).

%t a[n_] := n * GCD[5, n]; Array[a, 100] (* _Amiram Eldar_, Mar 13 2026 *)

%o (Python)

%o from math import gcd

%o def a(n): return n * gcd(5, n)

%o print([a(n) for n in range(1, 31)])

%o (PARI) a(n) = n*gcd(n, 5); \\ _Michel Marcus_, Mar 19 2026

%Y Cf. A109009, A203648 (d=3), A394223 (d=7).

%K nonn,mult,easy

%O 1,2

%A _Remco Havenaar_, Mar 13 2026