A019536 Number of length n necklaces with integer entries that cover an initial interval of positive integers.

1, 2, 5, 20, 109, 784, 6757, 68240, 787477, 10224812, 147512053, 2340964372, 40527565261, 760095929840, 15352212731933, 332228417657960, 7668868648772701, 188085259070219000, 4884294069438337429

Offset

1,2

Comments

Original name: a(n) = number of necklaces of n beads with up to n unlabeled colors.

The Moebius transform of this sequence is A060223.

Links

G. C. Greubel, Table of n, a(n) for n = 1..420

M. Goebel, On the number of special permutation-invariant orbits and terms, in: Applicable Algebra in Engin., Comm. and Comp. (AAECC 8), Volume 8, Number 6, 1997, pp. 505-509 (Lect. Notes Comp. Sci.); see p. 509 (stated as an open problem).

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]

Eric Weisstein's world of Mathematics, Necklaces.

Formula

See Mathematica code.

a(n) ~ (n-1)! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Jul 21 2019

From Petros Hadjicostas, Aug 19 2019: (Start)

The first formula is due to Philippe Deléham from the Crossrefs (see also the programs below). The second one follows easily from the first one. The third one follows from the second one using the associative property of Dirichlet convolutions.

a(n) = Sum_{k = 1..n} (k!/n) * Sum_{d|n} phi(d) * S2(n/d, k), where S2(n, k) = Stirling numbers of 2nd kind (A008277).

a(n) = (1/n) * Sum_{d|n} phi(d) * A000670(n/d).

a(n) = Sum_{d|n} A060223(d).

(End)

From Richard L. Ollerton, May 07 2021: (Start)

a(n) = (1/n)*Sum_{k=1..n} A000670(gcd(n,k)).

a(n) = (1/n)*Sum_{k=1..n} A000670(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

Example

a(3) = 5 since there are the following length 3 words up to rotation:
     111,  112, 122, 123, 132.
a(4) = 20 since there are the following length 4 words up to rotation:
     1111,
     1112, 1122, 1212, 1222,
     1123, 1132, 1213, 1223, 1232, 1233, 1322, 1323, 1332,
     1234, 1243, 1324, 1342, 1423, 1432.

Mathematica

Needs["DiscreteMath`Combinatorica`"];
mult[li:{__Integer}] := Multinomial @@ Length /@ Split[Sort[li]];
neck[li:{__Integer}] := Module[{n, d}, n=Plus @@ li; d=n-First[li]; Fold[ #1+(EulerPhi[ #2]*(n/#2)!)/Times @@ ((li/#2)!)&, 0, Divisors[GCD @@ li]]/n];
Table[(mult /@ Partitions[n]).(neck /@ Partitions[n]), {n, 24}]
(* second program: *)
a[n_] := Sum[DivisorSum[n, EulerPhi[#]*StirlingS2[n/#, k] k! &]/n, {k, 1, n}];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Mar 31 2016, after Philippe Deléham *)

Prog

(PARI) a(n) = sum(k=1, n, sumdiv(n, d, eulerphi(d)*stirling(n/d, k, 2)*k!)/n); \\ Michel Marcus, Mar 31 2016

Crossrefs

Cf. A000670, A008277, A019537, A060223.

Row sums of A087854. - Philippe Deléham

Sequence in context: A296727 A201224 A305922 * A129949 A127065 A168357

Adjacent sequences: A019533 A019534 A019535 * A019537 A019538 A019539

Keyword

easy,nonn

Author

Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de)

Extensions

Edited by Wouter Meeussen, Aug 06 2002

Corrected by T. D. Noe, Oct 31 2006

Edited by Andrew Howroyd, Aug 19 2019

Status

approved