A305150 Number of factorizations of n into distinct, pairwise indivisible factors greater than 1.

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, 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, 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

Offset

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

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

Formula

a(n) <= A045778(n) <= A001055(n). - Antti Karttunen, Dec 06 2018

Example

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).

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

Prog

(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

Crossrefs

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

Sequence in context: A294893 A336570 A121382 * A351202 A333145 A335449

Adjacent sequences: A305147 A305148 A305149 * A305151 A305152 A305153

Keyword

nonn

Author

Gus Wiseman, May 26 2018

Extensions

More terms from Antti Karttunen, Dec 06 2018

Status

approved