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A316365 Number of factorizations of n into factors > 1 such that every distinct subset of the factors has a different sum. 2
1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 4, 1, 6, 2, 2, 2, 9, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 9, 2, 4, 2, 4, 1, 6, 2, 7, 2, 2, 1, 10, 1, 2, 4, 9, 2, 5, 1, 4, 2, 4, 1, 14, 1, 2, 4, 4, 2, 5, 1, 11, 5, 2, 1, 9, 2, 2, 2, 7, 1, 10, 2, 4, 2, 2, 2, 15, 1, 4, 4, 9, 1, 5, 1, 7, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Also the number of factorizations of n into factors > 1 which form a knapsack partition.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10080

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

EXAMPLE

The a(24) = 7 factorizations are (2*2*2*3), (2*2*6), (2*3*4), (2*12), (3*8), (4*6), (24).

The a(54) = 6 factorizations are (2*3*3*3), (2*3*9), (2*27), (3*18), (6*9), (54).

MATHEMATICA

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@@Total/@Union[Subsets[#]]&]], {n, 100}]

PROG

(PARI)

primeprodbybits(v, b) = { my(m=1, i=1); while(b>0, if(b%2, m *= prime(v[i])); i++; b >>= 1); (m); };

sumbybits(v, b) = { my(s=0, i=1); while(b>0, s += (b%2)*v[i]; i++; b >>= 1); (s); };

all_distinct_subsets_have_different_sums(v) = { my(m=Map(), s, pp); for(i=0, (2^#v)-1, pp = primeprodbybits(v, i); s = sumbybits(v, i); if(mapisdefined(m, s), if(mapget(m, s)!=pp, return(0)), mapput(m, s, pp))); (1); };

A316365(n, m=n, facs=List([])) = if(1==n, all_distinct_subsets_have_different_sums(Vec(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs, d); s += A316365(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Oct 08 2018

CROSSREFS

Cf. A001055, A108917, A275972, A292886, A293627, A294150, A299702, A316313, A316314, A316364.

Sequence in context: A033273 A319685 A034836 * A292886 A317508 A323438

Adjacent sequences:  A316362 A316363 A316364 * A316366 A316367 A316368

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 30 2018

EXTENSIONS

More terms from Antti Karttunen, Oct 08 2018

STATUS

approved

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Last modified March 23 09:20 EDT 2019. Contains 321424 sequences. (Running on oeis4.)