A305149 Number of factorizations of n whose distinct factors are pairwise indivisible and greater than 1.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 6, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 8, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 6, 1, 2, 3, 3, 2, 5, 1, 5, 3, 2, 1, 8, 2, 2, 2, 4, 1, 8, 2, 3, 2, 2, 2, 6, 1, 3, 3, 6, 1, 5, 1, 4, 5

Offset: 1

Author

Gus Wiseman, May 26 2018

Keywords

• nonn

Examples

			The a(60) = 8 factorizations are (2*2*3*5), (2*2*15), (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).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Crossrefs

Cf. A001055, A001970, A007716, A034444, A045778, A259936, A281116, A285572, A302242, A303386, A303431, A305001, A305148, A305150.

Programs

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

PARI

pairwise_indivisible(v) = { for(i=1,#v,for(j=i+1,#v,if(!(v[j]%v[i]),return(0)))); (1); };
A305149(n, m=n, facs=List([])) = if(1==n, pairwise_indivisible(Set(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A305149(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Oct 08 2018

Extensions

More terms from Antti Karttunen, Oct 08 2018