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 A304716 Number of integer partitions of n whose distinct parts are connected. 67
 1, 2, 2, 3, 2, 5, 2, 6, 4, 9, 3, 15, 4, 18, 12, 25, 11, 41, 17, 54, 36, 72, 44, 113, 69, 145, 113, 204, 153, 302, 220, 394, 343, 541, 475, 771, 662, 1023, 968, 1398, 1314, 1929, 1822, 2566, 2565, 3440, 3446, 4677, 4688, 6187, 6407, 8216, 8544, 10975, 11436 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 FORMULA For n > 1, a(n) = A218970(n) + 1. - Gus Wiseman, Dec 04 2018 EXAMPLE The a(12) = 15 connected integer partitions and their corresponding connected multiset multisystems (see A112798, A302242) are the following.                      (12): {{1,1,2}}                     (6 6): {{1,2},{1,2}}                     (8 4): {{1,1,1},{1,1}}                     (9 3): {{2,2},{2}}                    (10 2): {{1,3},{1}}                   (4 4 4): {{1,1},{1,1},{1,1}}                   (6 3 3): {{1,2},{2},{2}}                   (6 4 2): {{1,2},{1,1},{1}}                   (8 2 2): {{1,1,1},{1},{1}}                 (3 3 3 3): {{2},{2},{2},{2}}                 (4 4 2 2): {{1,1},{1,1},{1},{1}}                 (6 2 2 2): {{1,2},{1},{1},{1}}               (4 2 2 2 2): {{1,1},{1},{1},{1},{1}}             (2 2 2 2 2 2): {{1},{1},{1},{1},{1},{1}} (1 1 1 1 1 1 1 1 1 1 1 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], Length[zsm[Union[#]]]===1&]], {n, 30}] CROSSREFS Cf. A000009, A003963, A048143, A054921, A218970, A285572, A286518, A302242, A304714, A305078, A305079, A322306, A322307. Sequence in context: A066656 A164896 A298422 * A237984 A118136 A258567 Adjacent sequences:  A304713 A304714 A304715 * A304717 A304718 A304719 KEYWORD nonn AUTHOR Gus Wiseman, May 17 2018 EXTENSIONS Name changed to distinguish from A218970 by Gus Wiseman, Dec 04 2018 STATUS approved

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Last modified December 15 00:30 EST 2019. Contains 329988 sequences. (Running on oeis4.)