A276548 Number of primitive (period n) bracelet structures using an infinite alphabet.

1, 1, 2, 5, 11, 33, 92, 347, 1347, 6338, 31949, 179265, 1071264, 6845487, 46162569, 327731596, 2437753739, 18948597836, 153498350744, 1293123237572, 11306475314372, 102425554267565, 959826755336241, 9290811905211847

Offset: 1

Author

Andrew Howroyd, Apr 09 2017

Keywords

• nonn

Crossrefs

Row sums of A276543.

Cf. A084708.

Programs

Mathematica

u[0, ] = 1; u[k, j_] := u[k, j] = Sum[Binomial[k - 1, i - 1] Total[u[k - i, j] #^(i - 1) & /@ Divisors[j]], {i, k}];
b[n_] := 1/n*Total[EulerPhi[#] u[Quotient[n, #], #]& /@ Divisors[n] ];
A084708[n_] := b[n]/2 + If[EvenQ[n], u[n/2, 2], Sum[Binomial[n/2 - 1/2, k] u[k, 2], {k, 0, n/2 - 1/2}]]/2;
a[n_] := Sum[MoebiusMu[n/d]*A084708[d], {d, Divisors[n]}];
Array[a, 24] (* Jean-François Alcover, Dec 28 2017, after Andrew Howroyd and Wouter Meeussen *)

Formula

a(n) = Sum_{d|n} mu(n/d) * A084708(d).