A318749 Number of pairwise relatively nonprime strict factorizations of n (no two factors are coprime).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 5, 2, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 7, 1, 2, 2, 3, 1, 1, 1, 3, 1

Offset

1,8

Comments

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Oct 08 2018

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

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

Index entries for sequences computed from exponents in factorization of n

Example

The a(96) = 7 factorizations are (96), (2*48), (4*24), (6*16), (8*12), (2*4*12), (2*6*8).
The a(480) = 18 factorizations:
  (480)
  (2*240) (4*120) (6*80) (8*60) (10*48) (12*40) (16*30) (20*24)
  (2*4*60) (2*6*40) (2*8*30) (2*10*24) (2*12*20) (4*6*20) (4*10*12) (6*8*10)
  (2*4*6*10)

Mathematica

strfacs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@Select[strfacs[n/d], Min@@#1>d&], {d, Rest[Divisors[n]]}]];
Table[Length[Select[strfacs[n], And@@(GCD[##]>1&)@@@Select[Tuples[#, 2], Less@@#&]&]], {n, 50}]

Prog

(PARI) A318749(n, m=n, facs=List([])) = if(1==n, (1!=gcd(Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs, d); s += A318749(n/d, d-1, newfacs))); (s)); \\ Antti Karttunen, Oct 08 2018

Crossrefs

Cf. A001055, A045778, A051185, A281116, A303282, A305843, A305854, A317748, A318715, A318717, A318721.

Sequence in context: A361566 A354991 A354990 * A347708 A050330 A339890

Adjacent sequences: A318746 A318747 A318748 * A318750 A318751 A318752

Keyword

nonn

Author

Gus Wiseman, Sep 02 2018

Extensions

More terms from Antti Karttunen, Oct 08 2018

Status

approved