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%I #9 Sep 23 2018 21:26:30
%S 1,1,1,1,1,2,1,2,1,2,1,3,1,2,2,2,1,3,1,3,2,2,1,4,1,2,2,3,1,5,1,3,2,2,
%T 2,5,1,2,2,5,1,5,1,3,3,2,1,6,1,3,2,3,1,5,2,5,2,2,1,8,1,2,3,4,2,5,1,3,
%U 2,5,1,9,1,2,3,3,2,5,1,6,2,2,1,9,2,2,2,5,1,9,2,3,2,2,2,10,1,3,3,5,1,5,1,5,4
%N Number of factorizations of n into factors > 1 such that every distinct submultiset of the factors has a different average.
%C Note that such a factorization is necessarily strict.
%H Antti Karttunen, <a href="/A316364/b316364.txt">Table of n, a(n) for n = 1..65537</a>
%e The a(80) = 6 factorizations are (80), (10*8), (16*5), (20*4), (40*2), (10*4*2).
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t Table[Length[Select[facs[n],UnsameQ@@Mean/@Union[Subsets[#]]&]],{n,50}]
%o (PARI)
%o choosebybits(v,m) = { my(s=vector(hammingweight(m)),i=j=1); while(m>0,if(m%2,s[j] = v[i];j++); i++; m >>= 1); s; };
%o hasdupavgs(v) = { my(avgs=Map(), k); for(i=1,(2^(#v))-1,k = (vecsum(choosebybits(v,i))/hammingweight(i)); if(mapisdefined(avgs,k),return(i),mapput(avgs,k,i))); (0); };
%o A316364(n, m=n, facs=List([])) = if(1==n, (0==hasdupavgs(Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A316364(n/d, d, newfacs))); (s)); \\ _Antti Karttunen_, Sep 21 2018
%Y Cf. A001055, A108917, A275972, A276024, A284640, A292886, A293627, A294150, A316313, A316314, A316365.
%K nonn
%O 1,6
%A _Gus Wiseman_, Jun 30 2018
%E More terms from _Antti Karttunen_, Sep 21 2018
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