A318746 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and successive parts (including the last with the first part) being indivisible.

1, 1, 1, 1, 2, 1, 3, 2, 4, 5, 6, 8, 11, 17, 20, 29, 41, 56, 79, 107, 155, 214, 305, 422, 604, 850, 1207, 1709, 2424, 3439, 4905, 6972, 9949, 14171, 20268, 28915, 41392, 59176, 84790, 121428, 174163, 249760, 358578, 514873, 739910, 1063523, 1529767, 2200926

Offset

1,5

Links

Andrew Howroyd, Table of n, a(n) for n = 1..100

Example

The a(14) = 17 Lyndon compositions with successive parts indivisible:
  (14)
  (3,11) (4,10) (5,9) (6,8)
  (2,3,9) (2,5,7) (2,7,5) (3,4,7) (3,6,5) (3,7,4)
  (2,3,2,7) (2,3,4,5) (2,4,3,5) (2,4,5,3) (2,5,4,3)
  (2,3,2,4,3)

Mathematica

LyndonQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And]&&Array[RotateRight[q, #]&, Length[q], 1, UnsameQ];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Or[Length[#]==1, LyndonQ[#]&&And@@Not/@Divisible@@@Partition[#, 2, 1, 1]]&]], {n, 20}]

Prog

(PARI)
b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i, j)->i%j<>0))); vector(n, n, 1 + sumdiv(n, d, moebius(d)*v[n/d])/n)} \\ Andrew Howroyd, Nov 01 2019

Crossrefs

Cf. A000740, A008965, A059966, A285573, A303362, A318726, A318727, A318729, A318730, A318731, A318745, A318747.

Sequence in context: A022447 A117194 A340647 * A318729 A024467 A215489

Adjacent sequences: A318743 A318744 A318745 * A318747 A318748 A318749

Keyword

nonn

Author

Gus Wiseman, Sep 02 2018

Extensions

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018

Status

approved