A304714 - OEIS

A304714

Number of connected strict integer partitions of n.

1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 5, 2, 5, 5, 6, 5, 10, 6, 12, 12, 13, 14, 21, 17, 23, 26, 30, 31, 46, 38, 51, 55, 61, 70, 87, 85, 102, 116, 128, 138, 171, 169, 204, 225, 245, 272, 319, 334, 383, 429, 464, 515, 593, 629, 715, 790, 861, 950, 1082

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

COMMENTS

Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.

LINKS

Table of n, a(n) for n=1..60.

EXAMPLE

The a(19) = 6 strict integer partitions are (19), (9,6,4), (10,5,4), (10,6,3), (12,4,3), (8,6,3,2). Taking the normalized prime factors of each part (see A112798, A302242), we have the following connected multiset multisystems.

(19): {{8}}

(9,6,4): {{2,2},{1,2},{1,1}}

(10,5,4): {{1,3},{3},{1,1}}

(10,6,3): {{1,3},{1,2},{2}}

(12,4,3): {{1,1,2},{1,1},{2}}

(8,6,3,2): {{1,1,1},{1,2},{2},{1}}

MATHEMATICA

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c==={}, s, zsm[Union[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[zsm[#]]===1&]], {n, 60}]

CROSSREFS

The Heinz numbers of these partitions are given by A328513.

Cf. A000009, A003963, A048143, A054921, A076078, A259936, A281116, A285572, A285573, A286518, A286520, A302242.

Sequence in context: A241316 A241312 A075989 * A085432 A029169 A202090

Adjacent sequences: A304711 A304712 A304713 * A304715 A304716 A304717

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 17 2018

STATUS

approved