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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 16 21:28 EDT 2021. Contains 347473 sequences. (Running on oeis4.)