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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.

%I A318746 #11 Nov 01 2019 22:19:23
%S A318746 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,
%T A318746 850,1207,1709,2424,3439,4905,6972,9949,14171,20268,28915,41392,59176,
%U A318746 84790,121428,174163,249760,358578,514873,739910,1063523,1529767,2200926
%N 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.
%H A318746 Andrew Howroyd, <a href="/A318746/b318746.txt">Table of n, a(n) for n = 1..100</a>
%e A318746 The a(14) = 17 Lyndon compositions with successive parts indivisible:
%e A318746   (14)
%e A318746   (3,11) (4,10) (5,9) (6,8)
%e A318746   (2,3,9) (2,5,7) (2,7,5) (3,4,7) (3,6,5) (3,7,4)
%e A318746   (2,3,2,7) (2,3,4,5) (2,4,3,5) (2,4,5,3) (2,5,4,3)
%e A318746   (2,3,2,4,3)
%t A318746 LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
%t A318746 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,LyndonQ[#]&&And@@Not/@Divisible@@@Partition[#,2,1,1]]&]],{n,20}]
%o A318746 (PARI)
%o A318746 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, ]}
%o A318746 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
%Y A318746 Cf. A000740, A008965, A059966, A285573, A303362, A318726, A318727, A318729, A318730, A318731, A318745, A318747.
%K A318746 nonn
%O A318746 1,5
%A A318746 _Gus Wiseman_, Sep 02 2018
%E A318746 Terms a(21) and beyond from _Andrew Howroyd_, Sep 08 2018