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A322367 Number of disconnected or empty integer partitions of n. 3
1, 0, 1, 2, 3, 6, 7, 14, 17, 27, 34, 54, 63, 98, 118, 165, 207, 287, 345, 474, 574, 757, 931, 1212, 1463, 1890, 2292, 2898, 3515, 4413, 5303 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

An integer partition is connected if the prime factorizations of its parts form a connected hypergraph. It is disconnected if it can be separated into two or more integer partitions with relatively prime products. For example, the integer partition (654321) has three connected components: (6432)(5)(1).

LINKS

Table of n, a(n) for n=0..30.

EXAMPLE

The a(3) = 2 through a(9) = 27 disconnected integer partitions:

  (21)   (31)    (32)     (51)      (43)       (53)        (54)

  (111)  (211)   (41)     (321)     (52)       (71)        (72)

         (1111)  (221)    (411)     (61)       (332)       (81)

                 (311)    (2211)    (322)      (431)       (432)

                 (2111)   (3111)    (331)      (521)       (441)

                 (11111)  (21111)   (421)      (611)       (522)

                          (111111)  (511)      (3221)      (531)

                                    (2221)     (3311)      (621)

                                    (3211)     (4211)      (711)

                                    (4111)     (5111)      (3222)

                                    (22111)    (22211)     (3321)

                                    (31111)    (32111)     (4221)

                                    (211111)   (41111)     (4311)

                                    (1111111)  (221111)    (5211)

                                               (311111)    (6111)

                                               (2111111)   (22221)

                                               (11111111)  (32211)

                                                           (33111)

                                                           (42111)

                                                           (51111)

                                                           (222111)

                                                           (321111)

                                                           (411111)

                                                           (2211111)

                                                           (3111111)

                                                           (21111111)

                                                           (111111111)

MATHEMATICA

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

Table[Length[Select[IntegerPartitions[n], Length[zsm[#]]!=1&]], {n, 20}]

CROSSREFS

Cf. A054921, A218970, A286518, A322335, A304714, A304716, A305078, A305079, A322306, A322307, A322337, A322338, A322368, A322369.

Sequence in context: A191615 A018606 A117087 * A319811 A000837 A200144

Adjacent sequences:  A322364 A322365 A322366 * A322368 A322369 A322370

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Dec 04 2018

STATUS

approved

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Last modified April 24 20:29 EDT 2019. Contains 322446 sequences. (Running on oeis4.)