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A321469 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.

%I A321469 #23 Jan 20 2025 13:09:28
%S A321469 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 A321469 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 A321469 2,4,1,8,1,2,3,3,2,5,1,6,2,2,1,7,2,2,2,5,1,7,2,3,2,2,2,8,1,3,3,5,1,5,1,5,5
%N A321469 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 A321469 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 A321469 Antti Karttunen, <a href="/A321469/b321469.txt">Table of n, a(n) for n = 1..65537</a>
%H A321469 Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>
%H A321469 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%e A321469 The a(72) = 8 multiset partitions with distinct block-sums:
%e A321469     {{1,1,1,2,2}}
%e A321469    {{1},{1,1,2,2}}
%e A321469    {{2},{1,1,1,2}}
%e A321469    {{1,1},{1,2,2}}
%e A321469    {{1,2},{1,1,2}}
%e A321469    {{2,2},{1,1,1}}
%e A321469   {{1},{2},{1,1,2}}
%e A321469   {{1},{1,1},{2,2}}
%e A321469 Missing from this list are:
%e A321469     {{1},{1},{1,2,2}}
%e A321469     {{1},{1,2},{1,2}}
%e A321469     {{2},{2},{1,1,1}}
%e A321469     {{2},{1,1},{1,2}}
%e A321469    {{1},{1},{1},{2,2}}
%e A321469    {{1},{1},{2},{1,2}}
%e A321469    {{1},{2},{2},{1,1}}
%e A321469   {{1},{1},{1},{2},{2}}
%t A321469 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A321469 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A321469 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A321469 Table[Length[Select[mps[primeMS[n]],UnsameQ@@Sort[Total/@#]&]],{n,100}]
%o A321469 (PARI)
%o A321469 A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
%o A321469 all_have_different_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == #facs));
%o A321469 A321469(n, m=n, facs=List([])) = if(1==n, all_have_different_sum_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A321469(n/d, d, newfacs))); (s)); \\ _Antti Karttunen_, Jan 20 2025
%Y A321469 Cf. A001055, A056239, A275780, A317144, A321455, A322794, A326514, A326515, A326516, A326519, A326535, A371734.
%K A321469 nonn
%O A321469 1,6
%A A321469 _Gus Wiseman_, Nov 11 2018
%E A321469 Data section extended to a(105) by _Antti Karttunen_, Jan 20 2025