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 A333517 Number of set partitions of {1, 2, ..., n} such that, for any two numbers in the same part, one divides the other. 2
 1, 1, 2, 3, 7, 9, 25, 30, 78, 138, 342, 386, 1307, 1448, 3406, 6818, 18907, 20478, 65901, 70781, 213704, 397874, 885118, 939377, 3624495, 5034048, 11032794, 20966732, 59398560, 62307000, 225641196, 235937708, 682530590, 1183122260, 2540294162, 4026533578, 15943721982, 16555409210, 35301649136, 59929463166 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) <= A000110(n) for all n since a(n) counts set partitions. a(n) >= n for all n, since the singletons form a suitable partition and separately moving 1 into each set creates n - 1 new suitable partitions. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..48 (terms n = 0..45 from Giovanni Resta) EXAMPLE For n = 3, the suitable partitions are {{1}, {2}, {3}}, {{1, 3}, {2}}, and {{1, 2}, {3}}, so a(3) = 3. MAPLE g:= proc(n, s) `if`(n<2, 1+nops(s), b(n, sort(map(i-> `if`(isprime(i), 1, i), s)))) end: b:= proc(n, s) option remember; add(`if`(irem(s[j], n)>0, 0, g(n-1, subsop(j=n, s))), j=1..nops(s))+g(n-1, [s[], n]) end: a:= n-> g(n, []): seq(a(n), n=0..28); # Alois P. Heinz, Mar 26 2020 MATHEMATICA g[n_, s_] := If[n<2, 1+Length[s], b[n, Sort[If[PrimeQ[#], 1, #]& /@ s]]]; b[n_, s_] := b[n, s] = Sum[If[Mod[s[[j]], n] > 0, 0, g[n-1, ReplacePart[s, j -> n]]], {j, 1, Length[s]}] + g[n-1, Append[s, n]]; a[n_] := g[n, {}]; a /@ Range[0, 28] (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz *) CROSSREFS Cf. A000110, A320430. Sequence in context: A319911 A024541 A250267 * A207643 A205488 A299437 Adjacent sequences: A333514 A333515 A333516 * A333518 A333519 A333520 KEYWORD nonn AUTHOR Robert Dougherty-Bliss, Mar 26 2020 EXTENSIONS a(25)-a(39) from Alois P. Heinz, Mar 26 2020 STATUS approved

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Last modified March 26 06:47 EDT 2023. Contains 361529 sequences. (Running on oeis4.)