cp's OEIS Frontend

A331685 Number of tree-factorizations of Heinz numbers of integer partitions of n.

%I A331685 #9 Dec 15 2020 09:11:08
%S A331685 1,3,7,23,69,261,943,3815,15107,63219,262791,1130953,4838813,21185125,
%T A331685 92593943,411160627,1823656199,8186105099,36728532951,166310761655
%N A331685 Number of tree-factorizations of Heinz numbers of integer partitions of n.
%C A331685 A tree-factorization of n > 1 is either (case 1) the number n itself, or (case 2) a sequence of two or more tree-factorizations, one of each part of a weakly increasing factorization of n into factors > 1.
%C A331685 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%F A331685 a(n) = Sum_i A281118(A215366(n,i)).
%e A331685 The a(1) = 1 through a(4) = 23 tree-factorizations:
%e A331685   2  3      5          7
%e A331685      4      6          9
%e A331685      (2*2)  8          10
%e A331685             (2*3)      12
%e A331685             (2*4)      16
%e A331685             (2*2*2)    (2*5)
%e A331685             (2*(2*2))  (2*6)
%e A331685                        (2*8)
%e A331685                        (3*3)
%e A331685                        (3*4)
%e A331685                        (4*4)
%e A331685                        (2*2*3)
%e A331685                        (2*2*4)
%e A331685                        (2*2*2*2)
%e A331685                        (2*(2*3))
%e A331685                        ((2*2)*4)
%e A331685                        (2*(2*4))
%e A331685                        (3*(2*2))
%e A331685                        (4*(2*2))
%e A331685                        (2*(2*2*2))
%e A331685                        (2*2*(2*2))
%e A331685                        ((2*2)*(2*2))
%e A331685                        (2*(2*(2*2)))
%t A331685 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A331685 physemi[n_]:=Prepend[Join@@Table[Tuples[physemi/@f],{f,Select[facs[n],Length[#]>1&]}],n];
%t A331685 Table[Sum[Length[physemi[Times@@Prime/@m]],{m,IntegerPartitions[n]}],{n,8}]
%o A331685 (PARI) \\ here TF(n) is n terms of A281118 as vector.
%o A331685 TF(n)={my(v=vector(n), w=vector(n)); w[1]=v[1]=1; for(k=2, n, w[k]=v[k]+1; forstep(j=n\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j] += w[k]^e*v[i]))); w}
%o A331685 a(n)={my(v=[prod(i=1, #p, prime(p[i])) | p<-partitions(n)], tf=TF(vecmax(v))); sum(i=1, #v, tf[v[i]])} \\ _Andrew Howroyd_, Dec 09 2020
%Y A331685 The orderless version is A319312.
%Y A331685 Factorizations are A001055.
%Y A331685 P-trees are A196545.
%Y A331685 Twice-factorizations are A281113.
%Y A331685 Tree-factorizations are A281118.
%Y A331685 Enriched p-trees are A289501.
%Y A331685 Cf. A000081 A000669, A000720, A001678, A005804, A056239, A112798, A141268, A273873, A330465.
%K A331685 nonn,more
%O A331685 1,2
%A A331685 _Gus Wiseman_, Jan 31 2020
%E A331685 a(13)-a(20) from _Andrew Howroyd_, Dec 09 2020