A394222 a(n) = n * gcd(5, n).

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, 26, 27, 28, 29, 150, 31, 32, 33, 34, 175, 36, 37, 38, 39, 200, 41, 42, 43, 44, 225, 46, 47, 48, 49, 250, 51, 52, 53, 54, 275, 56, 57, 58, 59, 300, 61, 62, 63, 64, 325, 66, 67, 68, 69, 350

Offset

1,2

Comments

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).

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.

Links

Table of n, a(n) for n=1..70.

Formula

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

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

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

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

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

Example

a(5) = 5 * gcd(5,5) = 25.
a(10) = 10 * gcd(5,10) = 50.
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).

Mathematica

a[n_] := n * GCD[5, n]; Array[a, 100] (* Amiram Eldar, Mar 13 2026 *)

Prog

(Python)
from math import gcd
def a(n): return n * gcd(5, n)
print([a(n) for n in range(1, 31)])
(PARI) a(n) = n*gcd(n, 5); \\ Michel Marcus, Mar 19 2026

Crossrefs

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

Sequence in context: A000336 A287433 A235041 * A080613 A055006 A139050

Adjacent sequences: A394219 A394220 A394221 * A394223 A394224 A394225

Keyword

nonn,mult,easy

Author

Remco Havenaar, Mar 13 2026

Status

approved