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0, 0, 0, 0, 1, 0, 2, 1, 2, 5, 4, 7, 6, 13, 14, 20, 30, 38, 50, 68, 97, 132, 176, 253, 328, 470, 631, 901, 1229, 1709, 2369, 3269, 4590, 6383, 8897, 12428, 17251, 24229, 33782, 47404, 66253, 92859, 130141, 182468, 256261, 359675, 505230, 710058, 997952, 1404214

Andrew Howroyd, <a href="/A328601/b328601.txt">Table of n, a(n) for n = 1..200</a>

(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 && j%i<>0))); vector(n, n, sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 26 2019

nonn,more,new

Terms a(26) and beyond from Andrew Howroyd, Oct 26 2019

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A necklace composition of n (A008965) is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.

Necklace compositions are A008965.

Numbers whose binary indices have no circularly adjacent divisors or multiples are A328608.

Cf. A000740, A008965, A032153, A167606, A318748, A328171, A328460, A328593, A328598, A328602, A328603, A328608, A328609.

allocated for Gus WisemanNumber of necklace compositions of n with no part circularly followed by a divisor or a multiple.

0, 0, 0, 0, 1, 0, 2, 1, 2, 5, 4, 7, 6, 13, 14, 20, 30, 38, 50, 68, 97, 132, 176, 253, 328

1,7

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.

Circularity means the last part is followed by the first.

a(n) = A318730(n) - 1.

The a(5) = 1 through a(13) = 6 necklace compositions (empty column not shown):

(2,3) (2,5) (3,5) (2,7) (3,7) (2,9) (5,7) (4,9)

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

(2,3,5) (4,7) (2,7,3) (6,7)

(2,5,3) (5,6) (3,4,5) (2,11)

(2,3,2,3) (3,5,4) (3,10)

(2,3,2,5) (2,3,5,3)

(2,3,4,3)

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

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], neckQ[#]&&And@@Not/@Divisible@@@Partition[#, 2, 1, 1]&&And@@Not/@Divisible@@@Reverse/@Partition[#, 2, 1, 1]&]], {n, 10}]

The non-necklace version is A328599.

The case forbidding divisors only is A328600 or A318729 (with singletons).

The non-necklace, non-circular version is A328508.

The version for co-primality (instead of indivisibility) is A328597.

Necklace compositions are A008965.

Numbers whose binary indices have no circularly adjacent divisors or multiples are A328608.

Cf. A000740, A032153, A167606, A318748, A328171, A328460, A328593, A328598, A328602, A328603, A328609.

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Gus Wiseman, Oct 25 2019

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