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 A051839 Number of partitions of n with parts (with repetitions) forming a division lattice (i.e., closed under GCD and LCM). 4
 1, 2, 3, 5, 6, 10, 11, 16, 19, 26, 27, 41, 42, 55, 64, 81, 83, 113, 115, 149, 165, 197, 203, 266, 276, 329, 358, 429, 440, 553, 565, 672, 722, 832, 874, 1060, 1085, 1252, 1342, 1558, 1603, 1901, 1955, 2249, 2410, 2708, 2805, 3287, 3394, 3852, 4078, 4594, 4756, 5456, 5668, 6379, 6738, 7484, 7767, 8884 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS EXAMPLE For n=6, the only one of the 11 partitions of 6 that fails is [3,2,1]; so a(6) = 10. MAPLE with(combinat): ans := []: b := []: for n to 30 do p := partition(n): np := nops(p): nn := np: print(n); for i to np do ss := convert(p[i], set):s := convert(ss, list): ns := nops(s): t := true: for j to ns-1 do for k from j+1 to ns do if evalb(not(member(gcd(s[j], s[k]), s)) or not(member(lcm(s[j], s[k]), s))) then t := false: fi: od: od: if t=false then nn := nn-1:fi od: ans := [op(ans), [n, np, nn]]: b := [op(b), [nn]]: od: print(ans); print(b); save b, ans, bans; MATHEMATICA ok[partition_] := Module[{p = Flatten[ If[ Length[#] > 2, Take[#, 2], #] & /@ Split[partition]], m}, m = Length[p]; Do[ If[ ! MemberQ[p, GCD[p[[i]], p[[j]]]] || ! MemberQ[p, LCM[p[[i]], p[[j]]]], Return[False]], {i, 1, m-1}, {j, i+1, m}] =!= False]; a[n_] := Length[ Select[ IntegerPartitions[n], ok]]; Table[an = a[n]; Print[an]; an, {n, 1, 60}] (* Jean-François Alcover, Sep 21 2012 *) PROG (PARI) ok(v)=v=vecsort(Vec(v), , 8); for(i=if(v[1]==1, 2, 1), #v-1, for(j=i+1, #v, if(!vecsearch(v, gcd(v[i], v[j])) || !vecsearch(v, lcm(v[i], v[j])), return(0)))); 1 a(n)=my(P=partitions(n)); sum(i=1, #P, ok(P[i])) \\ Charles R Greathouse IV, Sep 21 2012 (Sage) def A051839(n):     def closed(P):         S = Set(iter(P))         for p in S.subsets(2):             if not lcm(p) in S or not gcd(p) in S: return false         return true     count = 0     for p in Partitions(n):         if closed(p): count += 1     return count # Peter Luschny, Sep 21 2012 CROSSREFS Sequence in context: A298363 A018396 A003238 * A130714 A130689 A024560 Adjacent sequences:  A051836 A051837 A051838 * A051840 A051841 A051842 KEYWORD nonn,nice AUTHOR John McKay (mckay(AT)cs.concordia.ca), Dec 13 1999 EXTENSIONS More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 05 2003 a(46)-a(60) from Charles R Greathouse IV, Sep 21 2012 STATUS approved

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Last modified March 19 11:10 EDT 2019. Contains 321329 sequences. (Running on oeis4.)