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Antti Karttunen, <a href="/A316364/b316364.txt">Table of n, a(n) for n = 1..65537</a>

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, 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, 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

(PARI)

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; };

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); };

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

More terms from Antti Karttunen, Sep 21 2018

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Cf. A001055, A108917, A275972, A276024, A284640, A292886, A293627, A294150, A316313, A316314, A316365.

allocated for Gus WisemanNumber of factorizations of n into factors > 1 such that every distinct submultiset of the factors has a different average.

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, 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, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 6, 2, 2, 1, 9, 2, 2, 2

1,6

Note that such a factorization is necessarily strict.

The a(80) = 6 factorizations are (80), (10*8), (16*5), (20*4), (40*2), (10*4*2).

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

Table[Length[Select[facs[n], UnsameQ@@Mean/@Union[Subsets[#]]&]], {n, 50}]

Cf. A001055, A108917, A275972, A276024, A284640, A292886, A293627, A294150, A316313, A316314.

allocated

nonn

Gus Wiseman, Jun 30 2018

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editing

allocated for Gus Wiseman

allocated

approved