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Number of factorizations of n into distinct, pairwise indivisible factors greater than 1.
11

%I #18 Dec 06 2018 16:33:37

%S 1,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,3,1,2,1,2,1,5,1,1,2,2,

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

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

%N Number of factorizations of n into distinct, pairwise indivisible factors greater than 1.

%H Antti Karttunen, <a href="/A305150/b305150.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A305150/a305150.txt">Data supplement: n, a(n) computed for n = 1..100000</a>

%F a(n) <= A045778(n) <= A001055(n). - _Antti Karttunen_, Dec 06 2018

%e The a(60) = 6 factorizations are (3 * 4 * 5), (3 * 20), (4 * 15), (5 * 12), (6 * 10), (60). Missing from this list are (2 * 3 * 10), (2 * 5 * 6), (2 * 30).

%t 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@@ # && Select[Tuples[Union[#], 2], UnsameQ@@ # && Divisible@@ # &] == {} &]], {n, 100}]

%o (PARI) A305150(n, m=n, facs=List([])) = if(1==n, 1, my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m)&&factorback(apply(x -> (x%d),Vec(facs))), newfacs = List(facs); listput(newfacs,d); s += A305150(n/d, d-1, newfacs))); (s)); \\ _Antti Karttunen_, Dec 06 2018

%Y Cf. A001055, A001970, A007716, A045778, A162247, A259936, A275024, A285572, A281113, A281116, A303386, A303431, A305001, A305148, A305149, A305253.

%K nonn

%O 1,6

%A _Gus Wiseman_, May 26 2018

%E More terms from _Antti Karttunen_, Dec 06 2018