A318729 - OEIS

%I #17 Nov 04 2019 09:20:26

%S 1,1,1,1,2,1,3,2,4,6,6,8,11,19,21,30,41,59,79,112,157,219,305,430,605,

%T 860,1210,1727,2424,3463,4905,7001,9954,14211,20271,28980,41392,59254,

%U 84800,121540,174163,249932,358578,515091,739933,1063827,1529767,2201383

%N Number of cyclic compositions (necklaces of positive integers) summing to n that have only one part or whose consecutive parts (including the last with first) are indivisible.

%H Andrew Howroyd, <a href="/A318729/b318729.txt">Table of n, a(n) for n = 1..100</a>

%F a(n) = A328600(n) + 1. - _Andrew Howroyd_, Oct 27 2019

%e The a(13) = 11 cyclic compositions with successive parts indivisible:

%e   (13)

%e   (2,11) (3,10) (4,9) (5,8) (6,7)

%e   (2,4,7) (2,6,5) (2,8,3) (3,6,4)

%e   (2,3,5,3)

%t neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,neckQ[#]&&And@@Not/@Divisible@@@Partition[#,2,1,1]]&]],{n,20}]

%o (PARI)

%o 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 seq(n)={my(v=sum(k=1, n, k*b(n, k, (i,j)->i%j<>0))); vector(n, n, 1 + sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ _Andrew Howroyd_, Oct 27 2019

%Y Cf. A000740, A008965, A059966, A167606, A285573, A303362, A304713, A316476, A318726, A318727, A318728, A318730, A328600.

%K nonn

%O 1,5

%A _Gus Wiseman_, Sep 02 2018

%E Terms a(21) and beyond from _Andrew Howroyd_, Sep 08 2018

%E Name corrected by _Gus Wiseman_, Nov 04 2019