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Number of factorizations of n into factors > 1 with different sums of prime indices. Number of multiset partitions of the multiset of prime indices of n with distinct block-sums.
20

%I #17 Apr 26 2021 09:13:14

%S 1,1,1,1,1,2,1,2,1,2,1,2,1,2,2,2,1,3,1,3,2,2,1,4,1,2,2,3,1,4,1,3,2,2,

%T 2,5,1,2,2,4,1,5,1,3,3,2,1,6,1,3,2,3,1,5,2,5,2,2,1,7,1,2,2,4,2,5,1,3,

%U 2,4,1,8,1,2,3,3,2,5,1,6,2,2,1,7,2,2,2

%N Number of factorizations of n into factors > 1 with different sums of prime indices. Number of multiset partitions of the multiset of prime indices of n with distinct block-sums.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum of prime indices of n is A056239(n).

%H Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>

%e The a(72) = 8 multiset partitions with distinct block-sums:

%e {{1,1,1,2,2}}

%e {{1},{1,1,2,2}}

%e {{2},{1,1,1,2}}

%e {{1,1},{1,2,2}}

%e {{1,2},{1,1,2}}

%e {{2,2},{1,1,1}}

%e {{1},{2},{1,1,2}}

%e {{1},{1,1},{2,2}}

%e Missing from this list are:

%e {{1},{1},{1,2,2}}

%e {{1},{1,2},{1,2}}

%e {{2},{2},{1,1,1}}

%e {{2},{1,1},{1,2}}

%e {{1},{1},{1},{2,2}}

%e {{1},{1},{2},{1,2}}

%e {{1},{2},{2},{1,1}}

%e {{1},{1},{1},{2},{2}}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t Table[Length[Select[mps[primeMS[n]],UnsameQ@@Sort[Total/@#]&]],{n,100}]

%Y Cf. A001055, A275780, A317144, A321455, A322794, A326514, A326515, A326516, A326519, A326535.

%K nonn

%O 1,6

%A _Gus Wiseman_, Nov 11 2018