A328602 Number of necklace compositions of n where no pair of circularly adjacent parts is relatively prime.

0, 1, 1, 2, 1, 4, 1, 5, 3, 8, 1, 16, 1, 20, 9, 35, 2, 69, 3, 111, 24, 190, 13, 384, 31, 646, 102, 1212, 113, 2348, 227, 4254, 613, 7993, 976, 15459, 1915, 28825, 4357, 54988, 7868, 105826, 15760, 201115, 33376, 385590, 63974, 744446, 128224, 1428047, 262914, 2754037

Offset: 1

Author

Gus Wiseman, Oct 25 2019

Keywords

• nonn

Comments

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.

Circularity means the last part is followed by the first.

Examples

			The a(2) = 1 through a(10) = 8 necklace compositions:
  (2)  (3)  (4)    (5)  (6)      (7)  (8)        (9)      (10)
            (2,2)       (2,4)         (2,6)      (3,6)    (2,8)
                        (3,3)         (4,4)      (3,3,3)  (4,6)
                        (2,2,2)       (2,2,4)             (5,5)
                                      (2,2,2,2)           (2,2,6)
                                                          (2,4,4)
                                                          (2,2,2,4)
                                                          (2,2,2,2,2)
The a(19) = 3 necklace compositions are: (19), (3,6,4,6), (2,2,6,3,6).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

The non-necklace, non-circular version is A178470.

The version for indivisibility (rather than co-primality) is A328600.

The circularly coprime (as opposed to anti-coprime) version is A328597.

Partitions with no consecutive parts relatively prime are A328187.

Cf. A000031, A000740, A008965, A032153, A318728, A318729, A318748, A328172, A328188, A328220, A328335, A328336, A328601, A328609.

Programs

Mathematica

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&And@@Not/@CoprimeQ@@@Partition[#,2,1,1]&]],{n,10}]

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)->gcd(i,j)<>1))); vector(n, n, sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 26 2019

Extensions

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