A339564 Number of ways to choose a distinct factor in a factorization of n (pointed factorizations).

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 3, 3, 7, 1, 7, 1, 7, 3, 3, 1, 14, 2, 3, 4, 7, 1, 10, 1, 12, 3, 3, 3, 17, 1, 3, 3, 14, 1, 10, 1, 7, 7, 3, 1, 26, 2, 7, 3, 7, 1, 14, 3, 14, 3, 3, 1, 25, 1, 3, 7, 19, 3, 10, 1, 7, 3, 10, 1, 36, 1, 3, 7, 7, 3, 10, 1, 26, 7, 3

Offset: 1

Author

Gus Wiseman, Apr 10 2021

Keywords

• nonn

Examples

			The pointed factorizations of n for n = 2, 4, 6, 8, 12, 24, 30:
  ((2))  ((4))    ((6))    ((8))      ((12))     ((24))       ((30))
         ((2)*2)  ((2)*3)  ((2)*4)    ((2)*6)    ((3)*8)      ((5)*6)
                  (2*(3))  (2*(4))    (2*(6))    (3*(8))      (5*(6))
                           ((2)*2*2)  ((3)*4)    ((4)*6)      ((2)*15)
                                      (3*(4))    (4*(6))      (2*(15))
                                      ((2)*2*3)  ((2)*12)     ((3)*10)
                                      (2*2*(3))  (2*(12))     (3*(10))
                                                 ((2)*2*6)    ((2)*3*5)
                                                 (2*2*(6))    (2*(3)*5)
                                                 ((2)*3*4)    (2*3*(5))
                                                 (2*(3)*4)
                                                 (2*3*(4))
                                                 ((2)*2*2*3)
                                                 (2*2*2*(3))
		

Crossrefs

The additive version is A000070 (strict: A015723).

The unpointed version is A001055 (strict: A045778, ordered: A074206, listed: A162247).

Allowing point (1) gives A057567.

Choosing a position instead of value gives A066637.

The ordered additive version is A336875.

A000005 counts divisors.

A001787 count normal multisets with a selected position.

A001792 counts compositions with a selected position.

A006128 counts partitions with a selected position.

A066186 count strongly normal multisets with a selected position.

A254577 counts ordered factorizations with a selected position.

Cf. A007716, A050336, A281113, A281116, A292886, A293627.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[Union[fac]],{fac,facs[n]}],{n,50}]

Formula

a(n) = A057567(n) - A001055(n).

a(n) = Sum_{d|n, d>1} A001055(n/d).