A329632 Number of connected integer partitions of n whose distinct parts are pairwise indivisible.

1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 6, 4, 6, 1, 9, 2, 10, 6, 13, 3, 15, 6, 18, 8, 22, 9, 29, 10, 30, 20, 40, 22, 48, 24, 57, 36, 68

Offset

0,5

Comments

Given an integer partition y of length k, let G(y) be the simple labeled graph with vertices {1..k} and edges between any two vertices i, j such that GCD(y_i, y_j) > 1. For example, G(6,14,15,35) is a 4-cycle. A partition y is said to be connected if G(y) is a connected graph.

Links

Table of n, a(n) for n=0..40.

Example

The a(n) partitions for n = 1, 4, 6, 10, 12, 14:
  (1)  (4)    (6)      (10)         (12)           (14)
       (2,2)  (3,3)    (5,5)        (6,6)          (7,7)
              (2,2,2)  (6,4)        (4,4,4)        (8,6)
                       (2,2,2,2,2)  (3,3,3,3)      (10,4)
                                    (2,2,2,2,2,2)  (6,4,4)
                                                   (2,2,2,2,2,2,2)

Mathematica

stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], GCD@@s[[#]]>1&]}, If[c=={}, s, zsm[Sort[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n], stableQ[#, Divisible]&&Length[zsm[#]]<=1&]], {n, 0, 30}]

Crossrefs

The Heinz numbers of these partitions are given by A329559.

The strict version is A304717.

Connected partitions are A218970.

Pairwise indivisible partitions are A305148.

Cf. A048143, A286518, A286520, A304714, A304716, A305078.

Sequence in context: A351465 A309634 A309633 * A014599 A274771 A075825

Adjacent sequences: A329629 A329630 A329631 * A329633 A329634 A329635

Keyword

nonn,more

Author

Gus Wiseman, Nov 18 2019

Status

approved