A328966 Number of strict factorizations of n with integer average.

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

Offset: 2

Author

Gus Wiseman, Nov 16 2019

Keywords

• nonn

Examples

			The a(n) factorizations for n = 2, 8, 24, 48, 96:
  (2)  (8)    (24)     (32)    (48)     (96)
       (2*4)  (4*6)    (4*8)   (6*8)    (2*48)
              (2*12)   (2*16)  (2*24)   (4*24)
              (2*3*4)          (4*12)   (6*16)
                               (2*4*6)  (8*12)
                                        (3*4*8)
                                        (2*3*16)
                                        (2*4*12)
		

Crossrefs

The non-strict version is A326622.

Partitions with integer average are A067538.

Strict partitions with integer average are A102627.

Heinz numbers of partitions with integer average are A316413.

Factorizations with integer geometric mean are A326028.

Cf. A001055, A051293, A078174, A078175, A326515, A326567/A326568, A326619/A326620, A326621, A326625.

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],UnsameQ@@#&&IntegerQ[Mean[#]]&]],{n,2,100}]