A318729 -id:A318729 - cp's OEIS frontend

A318747 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and adjacent parts (including the last with the first part) being indivisible (either way).

1, 1, 1, 1, 2, 1, 3, 2, 3, 5, 5, 8, 7, 12, 14, 20, 31, 37, 51, 64, 96, 129, 177, 246, 328, 465, 630, 889, 1230, 1692, 2370, 3250, 4587, 6354, 8895, 12384, 17252, 24180, 33777, 47336, 66254, 92752, 130142, 182337, 256246, 359500, 505231, 709787, 997951, 1403883

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(14) = 12 Lyndon compositions with adjacent parts indivisible either way:
  (14)
  (3,11) (4,10) (5,9) (6,8)
  (2,5,7) (2,7,5) (3,4,7) (3,7,4)
  (2,3,2,7) (2,3,4,5) (2,5,4,3)

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000740, A008965, A059966, A285573, A303362, A318726, A318727, A318729, A318730, A318731, A318745, A318746.

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,And[LyndonQ[#],And@@Not/@Divisible@@@Partition[#,2,1,1],And@@Not/@Divisible@@@Reverse/@Partition[#,2,1,1]]]&]],{n,20}]

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, 1 + sumdiv(n, d, moebius(d)*v[n/d])/n)} \\ Andrew Howroyd, Nov 01 2019

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018

A328674 Numbers whose distinct prime indices have no consecutive divisible parts.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 64, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 113, 119, 121, 123, 125, 127, 128, 131, 135

Offset: 1

Gus Wiseman, Oct 29 2019

nonn

First differs from A316476 in having 105, with prime indices {2, 3, 4}.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
For example, 45 is in the sequence because its distinct prime indices are {2,3} and 2 is not a divisor of 3.

These are the Heinz numbers of the partitions counted by A328675.
The strict version is A328603.
Partitions without consecutive divisibilities are A328171.
Compositions without consecutive divisibilities are A328460.
Cf. A056239, A112798, A316476, A318729, A328335, A328336, A328593, A328598.

Select[Range[100],!MatchQ[PrimePi/@First/@FactorInteger[#],{_,x_,y_,_}/;Divisible[y,x]]&]

cp's OEIS Frontend

A318747 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and adjacent parts (including the last with the first part) being indivisible (either way).

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