cp's OEIS Frontend

A050345 Number of ways to factor n into distinct factors with one level of parentheses.

%I A050345 #24 Apr 28 2025 21:34:41
%S A050345 1,1,1,1,1,3,1,3,1,3,1,6,1,3,3,4,1,6,1,6,3,3,1,13,1,3,3,6,1,12,1,7,3,
%T A050345 3,3,15,1,3,3,13,1,12,1,6,6,3,1,25,1,6,3,6,1,13,3,13,3,3,1,31,1,3,6,
%U A050345 12,3,12,1,6,3,12,1,37,1,3,6,6,3,12,1,25,4,3,1,31,3,3,3,13,1,31,3,6,3,3
%N A050345 Number of ways to factor n into distinct factors with one level of parentheses.
%C A050345 First differs from A296120 at a(36) = 15, A296120(36) = 14. - _Gus Wiseman_, Apr 27 2025
%C A050345 Each "part" in parentheses is distinct from all others at the same level. Thus (3*2)*(2) is allowed but (3)*(2*2) and (3*2*2) are not.
%C A050345 a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).
%H A050345 R. J. Mathar, <a href="/A050345/b050345.txt">Table of n, a(n) for n = 1..2519</a>
%F A050345 Dirichlet g.f.: Product_{n>=2}(1+1/n^s)^A045778(n).
%F A050345 a(n) = A050346(A025487^(-1)(A046523(n))), where A025487^(-1) is the inverse with A025487^(-1)(A025487(n))=n. - _R. J. Mathar_, May 25 2017
%F A050345 a(n) = A050346(A101296(n)). - _Antti Karttunen_, May 25 2017
%e A050345 12 = (12) = (6*2) = (6)*(2) = (4*3) = (4)*(3) = (3*2)*(2).
%e A050345 From _Gus Wiseman_, Apr 26 2025: (Start)
%e A050345 This is the number of ways to partition a factorization of n (counted by A001055) into a set of sets. For example, the a(12) = 6 choices are:
%e A050345   {{2},{2,3}}
%e A050345   {{2},{6}}
%e A050345   {{3},{4}}
%e A050345   {{2,6}}
%e A050345   {{3,4}}
%e A050345   {{12}}
%e A050345 (End)
%t A050345 facs[n_]:=If[n<=1,{{}}, Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d, Rest[Divisors[n]]}]];
%t A050345 sps[{}]:={{}};sps[set:{i_,___}] := Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,___}];
%t A050345 mps[set_]:=Union[Sort[Sort /@ (#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
%t A050345 Table[Sum[Length[Select[mps[y], UnsameQ@@#&&And@@UnsameQ@@@#&]], {y,facs[n]}],{n,30}] (* _Gus Wiseman_, Apr 26 2025 *)
%Y A050345 Cf. A050346-A050350. a(A002110)=A000258.
%Y A050345 For multisets of multisets we have A050336.
%Y A050345 For integer partitions we have a(p^k) = A050342(k), see A001970, A089259, A261049.
%Y A050345 For normal multiset partitions see A116539, A292432, A292444, A381996, A382214, A382216.
%Y A050345 The case of a unique choice (positions of 1) is A166684.
%Y A050345 Twice-partitions of this type are counted by A358914, see A270995, A281113, A294788.
%Y A050345 For sets of multisets we have A383310 (distinct products A296118).
%Y A050345 For multisets of sets we have we have A383311, see A296119.
%Y A050345 A001055 counts factorizations, strict A045778.
%Y A050345 A050320 counts factorizations into squarefree numbers, distinct A050326.
%Y A050345 A302494 gives MM-numbers of sets of sets.
%Y A050345 A382077 counts partitions that can be partitioned into a sets of sets, ranks A382200.
%Y A050345 A382078 counts partitions that cannot be partitioned into a sets of sets, ranks A293243.
%Y A050345 Cf. A000009, A002865, A005117, A045782, A279785, A293511, A296120, A316439, A381441, A382201.
%K A050345 nonn
%O A050345 1,6
%A A050345 _Christian G. Bower_, Oct 15 1999